SlideShare uma empresa Scribd logo
1 de 57
Baixar para ler offline
Indice   Monomials   Adding and subtracting       Identities and Equations   Solving   Exercises




                 Algebraic expressions and equations




                             Matem´ticas 1o E.S.O.
                                  a
                               Alberto Pardo Milan´s
                                                  e




                                              -
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




          1 Monomials


          2 Adding and subtracting monomials


          3 Identities and Equations


          4 Solving


          5 Exercises




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




                                   Monomials




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Monomials


 What´s a monomial?

          A variable is a symbol.

          An algebraic expression in variables x, y, z, a, r, t . . . k is an
          expression constructed with the variables and numbers using
          addition, multiplication, and powers.

          A number multiplied with a variable in an algebraic expression is
          named coefficient.

          A product of positive integer powers of a fixed set of variables
          multiplied by some coefficient is called a monomial.
                           2 2 2 3
          Examples: 3x,      xy , x y z.
                           3

Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Monomials


 Like monomials and unlike monomials

          In a monomial with only one variable, the power is called its order,
          or sometimes its degree.
          Example: Deg(5x4 )=4.

          In a monomial with several variables, the order/degree is the sum
          of the powers.
          Example: Deg(x2 z 4 )=6.

          Monomials are called similar or like ones, if they are identical or
          differed only by coefficients.
                              2
          Example: 2x3 y 2 and x3 y 2 are like monomials. 4xy 2 and 4y 2 x4
                              5
          are unlike monomials.



Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




                        Adding and subtracting
                             monomials



Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials            Adding and subtracting     Identities and Equations   Solving   Exercises

 Adding and subtracting monomials


 Adding and Subtracting

          You can ONLY add or subtract like monomials.

          To add or subtract like monomials use the same rules as with
          integers.

          Example: 3x + 4x = (3 + 4)x = 7x.

          Example: 20a − 24a = (20 − 24)a = −4a.

          Example: 7x + 5y ⇐= you can´t add unlike monomials.




Alberto Pardo Milan´s
                   e                                         Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




                    Identities and Equations




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 What´s an equation? Identities vs equations.

          An equation is a mathematical expression stating that a pair of
          algebraic expression are the same.

          If the equation is true for every value of the variables then it´s
          called Identity.

          An identity is a mathematical relationship equating one quantity to
          another which may initially appear to be different.

          Example: x2 − x3 + x + 1 = 3x4 is an equation,
          3x2 − x + 1 = x2 − x + 2 + 2x2 − 1 is an identity.




Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 Parts of an equation.

          In an equation:
          the variables are named unknowns (or indeterminate quantities),
          the number multiplied with a variable is named coefficient, a
          term is a summand of the equation, the highest power of the
          unknowns is called the order/degree of the equation.

          Example: In the equation 2x3 + 4y + 1 = 4:
          the unknowns are x and y,
          the coefficient of x3 is 2
          and
          the coefficient of y is 4,
          the order of the equation is 3.



Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 Parts of an equation.

          In an equation:
          the variables are named unknowns (or indeterminate quantities),
          the number multiplied with a variable is named coefficient, a
          term is a summand of the equation, the highest power of the
          unknowns is called the order/degree of the equation.

          Example: In the equation 2x3 + 4y + 1 = 4:
          the unknowns are x and y,
          the coefficient of x3 is 2
          and
          the coefficient of y is 4,
          the order of the equation is 3.



Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 Parts of an equation.

          In an equation:
          the variables are named unknowns (or indeterminate quantities),
          the number multiplied with a variable is named coefficient, a
          term is a summand of the equation, the highest power of the
          unknowns is called the order/degree of the equation.

          Example: In the equation 2x3 + 4y + 1 = 4:
          the unknowns are x and y,
          the coefficient of x3 is 2
          and
          the coefficient of y is 4,
          the order of the equation is 3.



Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 Parts of an equation.

          In an equation:
          the variables are named unknowns (or indeterminate quantities),
          the number multiplied with a variable is named coefficient, a
          term is a summand of the equation, the highest power of the
          unknowns is called the order/degree of the equation.

          Example: In the equation 2x3 + 4y + 1 = 4:
          the unknowns are x and y,
          the coefficient of x3 is 2
          and
          the coefficient of y is 4,
          the order of the equation is 3.



Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials    Adding and subtracting     Identities and Equations   Solving   Exercises

 Identities and Equations


 Parts of an equation.

          In an equation:
          the variables are named unknowns (or indeterminate quantities),
          the number multiplied with a variable is named coefficient, a
          term is a summand of the equation, the highest power of the
          unknowns is called the order/degree of the equation.

          Example: In the equation 2x3 + 4y + 1 = 4:
          the unknowns are x and y,
          the coefficient of x3 is 2
          and
          the coefficient of y is 4,
          the order of the equation is 3.



Alberto Pardo Milan´s
                   e                                 Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




                                         Solving




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 Solution of an equation.

           You are solving a equation when you replace a variable with a
           value and the mathematical expressions are still the same. The
           value for the variables is the solution of the equation.

           Example: In the equation 2x = 10 the solution is 5, because
           2 · 5 = 10.

           Example: Sam is 9 years old. This is seven years younger than her
           sister Rose’s age. We can solve an equation to find Rose’s age:
           x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
           old.




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 Solution of an equation.

           You are solving a equation when you replace a variable with a
           value and the mathematical expressions are still the same. The
           value for the variables is the solution of the equation.

           Example: In the equation 2x = 10 the solution is 5, because
           2 · 5 = 10.

           Example: Sam is 9 years old. This is seven years younger than her
           sister Rose’s age. We can solve an equation to find Rose’s age:
           x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
           old.




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 Solution of an equation.

           You are solving a equation when you replace a variable with a
           value and the mathematical expressions are still the same. The
           value for the variables is the solution of the equation.

           Example: In the equation 2x = 10 the solution is 5, because
           2 · 5 = 10.

           Example: Sam is 9 years old. This is seven years younger than her
           sister Rose’s age. We can solve an equation to find Rose’s age:
           x − 7 = 9, the solution of the equation is 16, so Rose is 16 years
           old.




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           To solve equations you can use the balance method, you must
           carry out the same operations in both sides and in the same order.
           You must use these properties:

           • Addition Property of Equalities: If you add the same number to
           each side of an equation, the two sides remain equal (note you can
           also add negative numbers).
           Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2

           • Multiplication Property of Equalities: If you multiply by the same
           number each side of an equation, the two sides remain equal (note
           you can also multiply by fractions).
                     x              x
           Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
                      5             5

Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           To solve equations you can use the balance method, you must
           carry out the same operations in both sides and in the same order.
           You must use these properties:

           • Addition Property of Equalities: If you add the same number to
           each side of an equation, the two sides remain equal (note you can
           also add negative numbers).
           Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2

           • Multiplication Property of Equalities: If you multiply by the same
           number each side of an equation, the two sides remain equal (note
           you can also multiply by fractions).
                     x              x
           Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
                      5             5

Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           To solve equations you can use the balance method, you must
           carry out the same operations in both sides and in the same order.
           You must use these properties:

           • Addition Property of Equalities: If you add the same number to
           each side of an equation, the two sides remain equal (note you can
           also add negative numbers).
           Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2

           • Multiplication Property of Equalities: If you multiply by the same
           number each side of an equation, the two sides remain equal (note
           you can also multiply by fractions).
                     x              x
           Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
                      5             5

Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           To solve equations you can use the balance method, you must
           carry out the same operations in both sides and in the same order.
           You must use these properties:

           • Addition Property of Equalities: If you add the same number to
           each side of an equation, the two sides remain equal (note you can
           also add negative numbers).
           Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2

           • Multiplication Property of Equalities: If you multiply by the same
           number each side of an equation, the two sides remain equal (note
           you can also multiply by fractions).
                     x              x
           Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30
                      5             5

Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Solving


 The balance method.

           • Brackets: Sometimes you will need to solve equations involving
           brackets. If brackets appear, first remove the brackets by expanding
           each bracketed expression.
           Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒
                         2x    8
           2x = 8 =⇒        = =⇒ x = 4
                         2     2
           Use all three properties to solve equations:
           Example: Solve 4x + 3 · (x − 25) = 240:
           First we remove brackets: 3 · (x − 25) = 3x − 75 so
           4x+3x − 75 = 240.
           Them we use addition property:
           4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315.
                                                     7x   315
           Now we can use multiplication property:      =
                                                      7    7
           So the solution is x = 45.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises




                                      Exercises




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice          Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
             a   4x + 2 = 26
             b   5(2x − 1) = 7(9 − x)
                 x 2x
             c     +     =7
                 2    3
             d   19 + 4x = 9 − x
             e   3(2x + 1) = x − 2
                 x 3x       1
             f     −     =
                 5    10    5




Alberto Pardo Milan´s
                   e                                  Algebraic expressions and equations
Indice          Monomials   Adding and subtracting      Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
                                                      x 2x
             a   4x + 2 = 26                     c      +    =7
                                                      2   3
                 4x = 26 − 2                          3x    4x    42
                                                         +      =
                 4x = 24                               6     6    6
                 x = 24 : 4                           3x + 4x = 42
                 x=6                                  7x = 42
                                                      x = 42 : 7
             b   5(2x − 1) = 7(9 − x)
                                                      x=6
                 10x − 5 = 63 − 7x
                 10x + 7x = 63 + 5               d    19 + 4x = 9 − x
                 17x = 68                             4x + x = 9 − 19
                 x = 68 : 17                          5x = −10
                 x=4                                  x = −10 : 5
                                                      x = −2

Alberto Pardo Milan´s
                   e                                   Algebraic expressions and equations
Indice          Monomials   Adding and subtracting      Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
                                                      x 2x
             a   4x + 2 = 26                     c      +    =7
                                                      2   3
                 4x = 26 − 2                          3x    4x    42
                                                         +      =
                 4x = 24                               6     6    6
                 x = 24 : 4                           3x + 4x = 42
                 x=6                                  7x = 42
                                                      x = 42 : 7
             b   5(2x − 1) = 7(9 − x)
                                                      x=6
                 10x − 5 = 63 − 7x
                 10x + 7x = 63 + 5               d    19 + 4x = 9 − x
                 17x = 68                             4x + x = 9 − 19
                 x = 68 : 17                          5x = −10
                 x=4                                  x = −10 : 5
                                                      x = −2

Alberto Pardo Milan´s
                   e                                   Algebraic expressions and equations
Indice          Monomials   Adding and subtracting      Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
                                                      x 2x
             a   4x + 2 = 26                     c      +    =7
                                                      2   3
                 4x = 26 − 2                          3x    4x    42
                                                         +      =
                 4x = 24                               6     6    6
                 x = 24 : 4                           3x + 4x = 42
                 x=6                                  7x = 42
                                                      x = 42 : 7
             b   5(2x − 1) = 7(9 − x)
                                                      x=6
                 10x − 5 = 63 − 7x
                 10x + 7x = 63 + 5               d    19 + 4x = 9 − x
                 17x = 68                             4x + x = 9 − 19
                 x = 68 : 17                          5x = −10
                 x=4                                  x = −10 : 5
                                                      x = −2

Alberto Pardo Milan´s
                   e                                   Algebraic expressions and equations
Indice          Monomials   Adding and subtracting      Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
                                                      x 2x
             a   4x + 2 = 26                     c      +    =7
                                                      2   3
                 4x = 26 − 2                          3x    4x    42
                                                         +      =
                 4x = 24                               6     6    6
                 x = 24 : 4                           3x + 4x = 42
                 x=6                                  7x = 42
                                                      x = 42 : 7
             b   5(2x − 1) = 7(9 − x)
                                                      x=6
                 10x − 5 = 63 − 7x
                 10x + 7x = 63 + 5               d    19 + 4x = 9 − x
                 17x = 68                             4x + x = 9 − 19
                 x = 68 : 17                          5x = −10
                 x=4                                  x = −10 : 5
                                                      x = −2

Alberto Pardo Milan´s
                   e                                   Algebraic expressions and equations
Indice          Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
             e   3(2x + 1) = x − 2
                 6x + 3 = x − 2
                 6x − x = −2 − 3
                 5x = −5
                 x = −5 : 5
                 x = −1
                 x 3x       1
             f     −     =
                 5    10    5
                 2x     3x     2
                     −     =
                 10     10    10
                 2x − 3x = 2
                 −1x = 2
                 x = 2 : (−1)
                 x = −2
Alberto Pardo Milan´s
                   e                                  Algebraic expressions and equations
Indice          Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 1

          Solve the equations:
             e   3(2x + 1) = x − 2
                 6x + 3 = x − 2
                 6x − x = −2 − 3
                 5x = −5
                 x = −5 : 5
                 x = −1
                 x 3x       1
             f     −     =
                 5    10    5
                 2x     3x     2
                     −     =
                 10     10    10
                 2x − 3x = 2
                 −1x = 2
                 x = 2 : (−1)
                 x = −2
Alberto Pardo Milan´s
                   e                                  Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 2

          Find a number such that 2 less than three times the number is 10.




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 2

          Find a number such that 2 less than three times the number is 10.


             Data: Let x be the number.
             Three times the number is 3x.
             2 less than three times the number is (3x − 2) and this is
             10.
                   3x − 2 = 10
                     3x = 12
                    x = 12 : 3
                      x=4                       Answer: The number is
                                                x = 4.



Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 2

          Find a number such that 2 less than three times the number is 10.


             Data: Let x be the number.
             Three times the number is 3x.
             2 less than three times the number is (3x − 2) and this is
             10.
                   3x − 2 = 10
                     3x = 12
                    x = 12 : 3
                      x=4                       Answer: The number is
                                                x = 4.



Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 2

          Find a number such that 2 less than three times the number is 10.


             Data: Let x be the number.
             Three times the number is 3x.
             2 less than three times the number is (3x − 2) and this is
             10.
                   3x − 2 = 10
                     3x = 12
                    x = 12 : 3
                      x=4                       Answer: The number is
                                                x = 4.



Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 3

          Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
          pounds less than twice her husband’s money. How many pounds
          does Mr. Roberts have? How many pounds does Mrs. Roberts
          have?




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving         Exercises

 Exercises


 Exercise 3

          Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
          pounds less than twice her husband’s money. How many pounds
          does Mr. Roberts have? How many pounds does Mrs. Roberts
          have?
           Data: They have 370 pounds. Mr. Roberts has x pounds.
           Twice Mr. Roberts’ money is 2x pounds.
           Mrs. Roberts has (2x − 155) pounds.

          x+(2x−155) = 370
          x + 2x − 155 = 370
               3x = 525
              x = 525 : 3
                x = 175      Answer: Mr. Roberts has                                       175
           370 − 175 = 195   pounds. Mrs. Roberts has                                      195
                             pounds.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving         Exercises

 Exercises


 Exercise 3

          Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
          pounds less than twice her husband’s money. How many pounds
          does Mr. Roberts have? How many pounds does Mrs. Roberts
          have?
           Data: They have 370 pounds. Mr. Roberts has x pounds.
           Twice Mr. Roberts’ money is 2x pounds.
           Mrs. Roberts has (2x − 155) pounds.

          x+(2x−155) = 370
          x + 2x − 155 = 370
               3x = 525
              x = 525 : 3
                x = 175      Answer: Mr. Roberts has                                       175
           370 − 175 = 195   pounds. Mrs. Roberts has                                      195
                             pounds.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving         Exercises

 Exercises


 Exercise 3

          Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155
          pounds less than twice her husband’s money. How many pounds
          does Mr. Roberts have? How many pounds does Mrs. Roberts
          have?
           Data: They have 370 pounds. Mr. Roberts has x pounds.
           Twice Mr. Roberts’ money is 2x pounds.
           Mrs. Roberts has (2x − 155) pounds.

          x+(2x−155) = 370
          x + 2x − 155 = 370
               3x = 525
              x = 525 : 3
                x = 175      Answer: Mr. Roberts has                                       175
           370 − 175 = 195   pounds. Mrs. Roberts has                                      195
                             pounds.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 4

          The length of a room exceeds the width by 5 feet. The length of
          the four walls is 30 feet. Find the dimensions of the room.




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 4

          The length of a room exceeds the width by 5 feet. The length of
          the four walls is 30 feet. Find the dimensions of the room.
            Data: The width of the room is x feet.
            The length of the room is (x + 5) feet.
            The length of the four walls is x + (x + 5) + x + (x + 5) feet
            and this is 30 feet.

          x + (x + 5) + x + (x + 5) = 30
          x + x + 5 + x + x + 5 = 30
          x + x + x + x = 30 − 5 − 5
                     4x = 20
                    x = 20 : 4
                      x=5
                    5 + 5 = 10           Answer: The room is 5 feet
                                         long and 10 feet wide.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 4

          The length of a room exceeds the width by 5 feet. The length of
          the four walls is 30 feet. Find the dimensions of the room.
            Data: The width of the room is x feet.
            The length of the room is (x + 5) feet.
            The length of the four walls is x + (x + 5) + x + (x + 5) feet
            and this is 30 feet.

          x + (x + 5) + x + (x + 5) = 30
          x + x + 5 + x + x + 5 = 30
          x + x + x + x = 30 − 5 − 5
                     4x = 20
                    x = 20 : 4
                      x=5
                    5 + 5 = 10           Answer: The room is 5 feet
                                         long and 10 feet wide.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 4

          The length of a room exceeds the width by 5 feet. The length of
          the four walls is 30 feet. Find the dimensions of the room.
            Data: The width of the room is x feet.
            The length of the room is (x + 5) feet.
            The length of the four walls is x + (x + 5) + x + (x + 5) feet
            and this is 30 feet.

          x + (x + 5) + x + (x + 5) = 30
          x + x + 5 + x + x + 5 = 30
          x + x + x + x = 30 − 5 − 5
                     4x = 20
                    x = 20 : 4
                      x=5
                    5 + 5 = 10           Answer: The room is 5 feet
                                         long and 10 feet wide.
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 5

          Maria spent a third of her money on food. Then, she spent e21 on
          a present. At the end, she had the fifth of her money. How much
          money did she have at the beginning?




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 5

          Maria spent a third of her money on food. Then, she spent e21 on
          a present. At the end, she had the fifth of her money. How much
          money did she have at the beginning?
                                                                  x
           Data: At the beginig She had x euros. She spent          on food
                                                                  3
                                                           x
           and e21 on a present. At the end She had          euros.
                                                           5

                 x        x
             x − − 21 =                                                     7x = 315
                 3         5
             x   x    21    x                                             x = 315 : 7
               − −       =                                                    x = 45
             1   3     1    5
             15x    5x   315    3x
                 −     −      =
              15    15   15     15                  Answer: At the beginig
             15x − 5x − 315 = 3x                    She had e45.
             15x − 5x − 3x = 315
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 5

          Maria spent a third of her money on food. Then, she spent e21 on
          a present. At the end, she had the fifth of her money. How much
          money did she have at the beginning?
                                                                  x
           Data: At the beginig She had x euros. She spent          on food
                                                                  3
                                                           x
           and e21 on a present. At the end She had          euros.
                                                           5

                 x        x
             x − − 21 =                                                     7x = 315
                 3         5
             x   x    21    x                                             x = 315 : 7
               − −       =                                                    x = 45
             1   3     1    5
             15x    5x   315    3x
                 −     −      =
              15    15   15     15                  Answer: At the beginig
             15x − 5x − 315 = 3x                    She had e45.
             15x − 5x − 3x = 315
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 5

          Maria spent a third of her money on food. Then, she spent e21 on
          a present. At the end, she had the fifth of her money. How much
          money did she have at the beginning?
                                                                  x
           Data: At the beginig She had x euros. She spent          on food
                                                                  3
                                                           x
           and e21 on a present. At the end She had          euros.
                                                           5

                 x        x
             x − − 21 =                                                     7x = 315
                 3         5
             x   x    21    x                                             x = 315 : 7
               − −       =                                                    x = 45
             1   3     1    5
             15x    5x   315    3x
                 −     −      =
              15    15   15     15                  Answer: At the beginig
             15x − 5x − 315 = 3x                    She had e45.
             15x − 5x − 3x = 315
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 6

          John bought a book, a pencil and a notebook. The book cost the
          double of the notebook, and the pencil cost the fifth of the book
          and the notebook together. If he paid e18, what is the price of
          each article?




Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 6

          John bought a book, a pencil and a notebook. The book cost the
          double of the notebook, and the pencil cost the fifth of the book
          and the notebook together. If he paid e18, what is the price of
          each article?
           Data: The notebook cost x euros. The book cost 2x euros.
           The notebook and the book together cost (x + 2x) euros.
                              x + 2x
           The pencil cost           . He paid e18.
                                 5
                         x + 2x                 5x+10x+x+2x = 90
             x + 2x +            = 18
                           5                            18x = 90
            x     2x     x + 2x     18                 x = 90 : 18
               +      +           =
            1      1        5        1                    x=5
           5x     10x     x + 2x      90                 2x = 10
               +        +          =              x + 2x       15
            5       5         5        5                   =       = 3.
           5x + 10x + (x + 2x) = 90                   5         5
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 6
          Data: The notebook cost x euros. The book cost 2x euros.
          The notebook and the book together cost (x + 2x) euros.
                           x + 2x
          The pencil cost         . He paid e18.
                              5
                      x + 2x                5x+10x+x+2x = 90
            x + 2x +          = 18
                        5                          18x = 90
           x    2x    x + 2x     18              x = 90 : 18
              +     +          =
           1     1       5        1                  x=5
          5x    10x    x + 2x      90               2x = 10
              +      +          =             x + 2x     15
           5      5        5        5                 =     = 3.
          5x + 10x + (x + 2x) = 90               5        5


             Answer: The notebook cost e5, the book e10 and the
             pencil e3 .
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations
Indice        Monomials   Adding and subtracting     Identities and Equations   Solving   Exercises

 Exercises


 Exercise 6
          Data: The notebook cost x euros. The book cost 2x euros.
          The notebook and the book together cost (x + 2x) euros.
                           x + 2x
          The pencil cost         . He paid e18.
                              5
                      x + 2x                5x+10x+x+2x = 90
            x + 2x +          = 18
                        5                          18x = 90
           x    2x    x + 2x     18              x = 90 : 18
              +     +          =
           1     1       5        1                  x=5
          5x    10x    x + 2x      90               2x = 10
              +      +          =             x + 2x     15
           5      5        5        5                 =     = 3.
          5x + 10x + (x + 2x) = 90               5        5


             Answer: The notebook cost e5, the book e10 and the
             pencil e3 .
Alberto Pardo Milan´s
                   e                                Algebraic expressions and equations

Mais conteúdo relacionado

Mais procurados

Lecture 01 reals number system
Lecture 01 reals number systemLecture 01 reals number system
Lecture 01 reals number systemHazel Joy Chong
 
Variables & Expressions
Variables & ExpressionsVariables & Expressions
Variables & Expressionsrfant
 
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1Paso 2 profundizar y contextualizar el conocimiento de la unidad 1
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1luisfernando1371
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicasreyrey38
 
Order of Operations
Order of OperationsOrder of Operations
Order of Operationsdarrin4597
 
1.1/1.2 Properties of Real Numbers
1.1/1.2 Properties of Real Numbers1.1/1.2 Properties of Real Numbers
1.1/1.2 Properties of Real Numbersleblance
 
Solving One Step Inequalities
Solving One Step InequalitiesSolving One Step Inequalities
Solving One Step InequalitiesJessca Lundin
 
Y7 m280115workw ithnumb1
Y7 m280115workw ithnumb1Y7 m280115workw ithnumb1
Y7 m280115workw ithnumb13SNEducation
 
CBSE Class 10 Mathematics Real Numbers Topic
CBSE Class 10 Mathematics Real Numbers TopicCBSE Class 10 Mathematics Real Numbers Topic
CBSE Class 10 Mathematics Real Numbers TopicEdvie
 

Mais procurados (20)

Lecture 01 reals number system
Lecture 01 reals number systemLecture 01 reals number system
Lecture 01 reals number system
 
Fractions
FractionsFractions
Fractions
 
Variables & Expressions
Variables & ExpressionsVariables & Expressions
Variables & Expressions
 
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1Paso 2 profundizar y contextualizar el conocimiento de la unidad 1
Paso 2 profundizar y contextualizar el conocimiento de la unidad 1
 
Numeros reales
Numeros realesNumeros reales
Numeros reales
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicas
 
Ict
IctIct
Ict
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicas
 
Order of Operations
Order of OperationsOrder of Operations
Order of Operations
 
1.1/1.2 Properties of Real Numbers
1.1/1.2 Properties of Real Numbers1.1/1.2 Properties of Real Numbers
1.1/1.2 Properties of Real Numbers
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicas
 
Real number system
Real number systemReal number system
Real number system
 
Solving One Step Inequalities
Solving One Step InequalitiesSolving One Step Inequalities
Solving One Step Inequalities
 
Factorizacion clase 1
Factorizacion clase 1Factorizacion clase 1
Factorizacion clase 1
 
Rafsay castillo
Rafsay castilloRafsay castillo
Rafsay castillo
 
Y7 m280115workw ithnumb1
Y7 m280115workw ithnumb1Y7 m280115workw ithnumb1
Y7 m280115workw ithnumb1
 
Basic Algebra Ppt 1.1
Basic Algebra Ppt 1.1Basic Algebra Ppt 1.1
Basic Algebra Ppt 1.1
 
Chapter 4
Chapter 4Chapter 4
Chapter 4
 
CBSE Class 10 Mathematics Real Numbers Topic
CBSE Class 10 Mathematics Real Numbers TopicCBSE Class 10 Mathematics Real Numbers Topic
CBSE Class 10 Mathematics Real Numbers Topic
 
Inequalities
InequalitiesInequalities
Inequalities
 

Destaque (20)

MIT Math Syllabus 10-3 Lesson 2 : Polynomials
MIT Math Syllabus 10-3 Lesson 2 : PolynomialsMIT Math Syllabus 10-3 Lesson 2 : Polynomials
MIT Math Syllabus 10-3 Lesson 2 : Polynomials
 
Evaluating an Algebraic Expression
Evaluating an Algebraic ExpressionEvaluating an Algebraic Expression
Evaluating an Algebraic Expression
 
Poetry3
Poetry3Poetry3
Poetry3
 
Администрирование сайта
Администрирование сайтаАдминистрирование сайта
Администрирование сайта
 
16 april 2011
16 april 201116 april 2011
16 april 2011
 
10 Twitter tips
10 Twitter tips 10 Twitter tips
10 Twitter tips
 
debatavond over ondernemen in een social media wereld #AlechiaGoesWest
debatavond over ondernemen in een social media wereld #AlechiaGoesWest debatavond over ondernemen in een social media wereld #AlechiaGoesWest
debatavond over ondernemen in een social media wereld #AlechiaGoesWest
 
Team work ahmed adel
Team work ahmed adelTeam work ahmed adel
Team work ahmed adel
 
Nintendo Wii U 2013
Nintendo Wii U  2013 Nintendo Wii U  2013
Nintendo Wii U 2013
 
Portfolio
PortfolioPortfolio
Portfolio
 
Decimals
DecimalsDecimals
Decimals
 
20120927 voordracht bij marnixring waregem
20120927 voordracht bij marnixring waregem20120927 voordracht bij marnixring waregem
20120927 voordracht bij marnixring waregem
 
Database
DatabaseDatabase
Database
 
3 d tv
3 d tv3 d tv
3 d tv
 
20120922 social media tips & trics voor mycoachday lovendegem
20120922 social media tips & trics voor mycoachday lovendegem20120922 social media tips & trics voor mycoachday lovendegem
20120922 social media tips & trics voor mycoachday lovendegem
 
Indo us nuclear deal(nipun)
Indo us nuclear deal(nipun)Indo us nuclear deal(nipun)
Indo us nuclear deal(nipun)
 
Cose da mangiare gennaio 2014
Cose da mangiare gennaio 2014Cose da mangiare gennaio 2014
Cose da mangiare gennaio 2014
 
Unit9: Angles and Lines
Unit9: Angles and LinesUnit9: Angles and Lines
Unit9: Angles and Lines
 
Human right
Human rightHuman right
Human right
 
Black holes
Black holesBlack holes
Black holes
 

Semelhante a Unit6: Algebraic expressions and equations

Grade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd QuarterGrade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd Quarterjennytuazon01630
 
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptxJiyoona
 
ALGEBRAIC EXPRESSION.pptx
ALGEBRAIC EXPRESSION.pptxALGEBRAIC EXPRESSION.pptx
ALGEBRAIC EXPRESSION.pptxHoneyiaSipra
 
algebrappt1-160102122524 (8) (1).pptx
algebrappt1-160102122524 (8) (1).pptxalgebrappt1-160102122524 (8) (1).pptx
algebrappt1-160102122524 (8) (1).pptxNayanaLathiya
 
algebrappt1-160102122524 (8).pptx
algebrappt1-160102122524 (8).pptxalgebrappt1-160102122524 (8).pptx
algebrappt1-160102122524 (8).pptxNayanaLathiya
 
Class IX - Polynomials PPT
Class IX - Polynomials PPTClass IX - Polynomials PPT
Class IX - Polynomials PPTAlankritWadhwa
 
algebrappt1-160102122524 (8).pdf
algebrappt1-160102122524 (8).pdfalgebrappt1-160102122524 (8).pdf
algebrappt1-160102122524 (8).pdfNayanaLathiya
 
Algebra PPT
Algebra PPTAlgebra PPT
Algebra PPTsri_3007
 
Numeros reales franyuris rojas
Numeros reales franyuris rojasNumeros reales franyuris rojas
Numeros reales franyuris rojasFranyuris Rojas
 
Lesson 1 power point
Lesson 1 power pointLesson 1 power point
Lesson 1 power pointBeckyH13
 
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V KhanaparaPolynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V KhanaparaMD. G R Ahmed
 
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)ximenazuluaga3
 
Expresiones algebraicas factorizacion y radicalizacion
Expresiones algebraicas factorizacion y radicalizacionExpresiones algebraicas factorizacion y radicalizacion
Expresiones algebraicas factorizacion y radicalizacionCRISBELMARIADUMBARRI
 

Semelhante a Unit6: Algebraic expressions and equations (20)

Grade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd QuarterGrade 7 Mathematics Week 4 2nd Quarter
Grade 7 Mathematics Week 4 2nd Quarter
 
Presentacion
PresentacionPresentacion
Presentacion
 
Expresiones algebraicas
Expresiones algebraicas Expresiones algebraicas
Expresiones algebraicas
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicas
 
Expresiones algebraicas
Expresiones algebraicasExpresiones algebraicas
Expresiones algebraicas
 
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx
1-1-Slide-Show-Writing-and-Interpreting-Numerical-Expressions.pptx
 
ALGEBRAIC EXPRESSION.pptx
ALGEBRAIC EXPRESSION.pptxALGEBRAIC EXPRESSION.pptx
ALGEBRAIC EXPRESSION.pptx
 
algebrappt1-160102122524 (8) (1).pptx
algebrappt1-160102122524 (8) (1).pptxalgebrappt1-160102122524 (8) (1).pptx
algebrappt1-160102122524 (8) (1).pptx
 
algebrappt1-160102122524 (8).pptx
algebrappt1-160102122524 (8).pptxalgebrappt1-160102122524 (8).pptx
algebrappt1-160102122524 (8).pptx
 
9.1
9.19.1
9.1
 
Class IX - Polynomials PPT
Class IX - Polynomials PPTClass IX - Polynomials PPT
Class IX - Polynomials PPT
 
algebrappt1-160102122524 (8).pdf
algebrappt1-160102122524 (8).pdfalgebrappt1-160102122524 (8).pdf
algebrappt1-160102122524 (8).pdf
 
Algebra PPT
Algebra PPTAlgebra PPT
Algebra PPT
 
Numeros reales franyuris rojas
Numeros reales franyuris rojasNumeros reales franyuris rojas
Numeros reales franyuris rojas
 
Lesson 1 power point
Lesson 1 power pointLesson 1 power point
Lesson 1 power point
 
Algebraic terminology
Algebraic terminology Algebraic terminology
Algebraic terminology
 
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V KhanaparaPolynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
 
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)
 
Expresiones algebraicas factorizacion y radicalizacion
Expresiones algebraicas factorizacion y radicalizacionExpresiones algebraicas factorizacion y radicalizacion
Expresiones algebraicas factorizacion y radicalizacion
 
Math Algebra
Math AlgebraMath Algebra
Math Algebra
 

Mais de Alberto Pardo Milanés (20)

La francophonie
La francophonieLa francophonie
La francophonie
 
Proportions and percentages
Proportions and percentagesProportions and percentages
Proportions and percentages
 
Sexagesimal system
Sexagesimal systemSexagesimal system
Sexagesimal system
 
Handling data and probability
Handling data and probabilityHandling data and probability
Handling data and probability
 
Decimals
DecimalsDecimals
Decimals
 
Factors and multiples
Factors and multiplesFactors and multiples
Factors and multiples
 
Fractions and multiples
Fractions and multiplesFractions and multiples
Fractions and multiples
 
Counting Numbers
Counting NumbersCounting Numbers
Counting Numbers
 
The Race, INTEGERS MATHS GAME
The Race, INTEGERS MATHS GAMEThe Race, INTEGERS MATHS GAME
The Race, INTEGERS MATHS GAME
 
Unit00
Unit00Unit00
Unit00
 
Unit11
Unit11Unit11
Unit11
 
Unit12
Unit12Unit12
Unit12
 
Origami triangles
Origami trianglesOrigami triangles
Origami triangles
 
Hexagon
HexagonHexagon
Hexagon
 
Equilateraltriangle
EquilateraltriangleEquilateraltriangle
Equilateraltriangle
 
Pentagon
PentagonPentagon
Pentagon
 
Unit5
Unit5Unit5
Unit5
 
Unit4
Unit4Unit4
Unit4
 
Unit2
Unit2Unit2
Unit2
 
Unit3
Unit3Unit3
Unit3
 

Último

P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfP4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfYu Kanazawa / Osaka University
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 
Optical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxOptical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxPurva Nikam
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...CaraSkikne1
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSyedNadeemGillANi
 
3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptxmary850239
 
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustVani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustSavipriya Raghavendra
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsEugene Lysak
 
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.raviapr7
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRATanmoy Mishra
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.EnglishCEIPdeSigeiro
 
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTDR. SNEHA NAIR
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptxraviapr7
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
Protein Structure - threading Protein modelling pptx
Protein Structure - threading Protein modelling pptxProtein Structure - threading Protein modelling pptx
Protein Structure - threading Protein modelling pptxvidhisharma994099
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxAditiChauhan701637
 

Último (20)

P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdfP4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
P4C x ELT = P4ELT: Its Theoretical Background (Kanazawa, 2024 March).pdf
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 
Optical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxOptical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptx
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
 
3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx
 
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustVani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George Wells
 
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdfPersonal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
 
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
 
Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.Drug Information Services- DIC and Sources.
Drug Information Services- DIC and Sources.
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.
 
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
Protein Structure - threading Protein modelling pptx
Protein Structure - threading Protein modelling pptxProtein Structure - threading Protein modelling pptx
Protein Structure - threading Protein modelling pptx
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptx
 

Unit6: Algebraic expressions and equations

  • 1. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Algebraic expressions and equations Matem´ticas 1o E.S.O. a Alberto Pardo Milan´s e -
  • 2. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises 1 Monomials 2 Adding and subtracting monomials 3 Identities and Equations 4 Solving 5 Exercises Alberto Pardo Milan´s e Algebraic expressions and equations
  • 3. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Monomials Alberto Pardo Milan´s e Algebraic expressions and equations
  • 4. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Monomials What´s a monomial? A variable is a symbol. An algebraic expression in variables x, y, z, a, r, t . . . k is an expression constructed with the variables and numbers using addition, multiplication, and powers. A number multiplied with a variable in an algebraic expression is named coefficient. A product of positive integer powers of a fixed set of variables multiplied by some coefficient is called a monomial. 2 2 2 3 Examples: 3x, xy , x y z. 3 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 5. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Monomials Like monomials and unlike monomials In a monomial with only one variable, the power is called its order, or sometimes its degree. Example: Deg(5x4 )=4. In a monomial with several variables, the order/degree is the sum of the powers. Example: Deg(x2 z 4 )=6. Monomials are called similar or like ones, if they are identical or differed only by coefficients. 2 Example: 2x3 y 2 and x3 y 2 are like monomials. 4xy 2 and 4y 2 x4 5 are unlike monomials. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 6. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Adding and subtracting monomials Alberto Pardo Milan´s e Algebraic expressions and equations
  • 7. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Adding and subtracting monomials Adding and Subtracting You can ONLY add or subtract like monomials. To add or subtract like monomials use the same rules as with integers. Example: 3x + 4x = (3 + 4)x = 7x. Example: 20a − 24a = (20 − 24)a = −4a. Example: 7x + 5y ⇐= you can´t add unlike monomials. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 8. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Alberto Pardo Milan´s e Algebraic expressions and equations
  • 9. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations What´s an equation? Identities vs equations. An equation is a mathematical expression stating that a pair of algebraic expression are the same. If the equation is true for every value of the variables then it´s called Identity. An identity is a mathematical relationship equating one quantity to another which may initially appear to be different. Example: x2 − x3 + x + 1 = 3x4 is an equation, 3x2 − x + 1 = x2 − x + 2 + 2x2 − 1 is an identity. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 10. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Parts of an equation. In an equation: the variables are named unknowns (or indeterminate quantities), the number multiplied with a variable is named coefficient, a term is a summand of the equation, the highest power of the unknowns is called the order/degree of the equation. Example: In the equation 2x3 + 4y + 1 = 4: the unknowns are x and y, the coefficient of x3 is 2 and the coefficient of y is 4, the order of the equation is 3. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 11. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Parts of an equation. In an equation: the variables are named unknowns (or indeterminate quantities), the number multiplied with a variable is named coefficient, a term is a summand of the equation, the highest power of the unknowns is called the order/degree of the equation. Example: In the equation 2x3 + 4y + 1 = 4: the unknowns are x and y, the coefficient of x3 is 2 and the coefficient of y is 4, the order of the equation is 3. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 12. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Parts of an equation. In an equation: the variables are named unknowns (or indeterminate quantities), the number multiplied with a variable is named coefficient, a term is a summand of the equation, the highest power of the unknowns is called the order/degree of the equation. Example: In the equation 2x3 + 4y + 1 = 4: the unknowns are x and y, the coefficient of x3 is 2 and the coefficient of y is 4, the order of the equation is 3. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 13. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Parts of an equation. In an equation: the variables are named unknowns (or indeterminate quantities), the number multiplied with a variable is named coefficient, a term is a summand of the equation, the highest power of the unknowns is called the order/degree of the equation. Example: In the equation 2x3 + 4y + 1 = 4: the unknowns are x and y, the coefficient of x3 is 2 and the coefficient of y is 4, the order of the equation is 3. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 14. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Identities and Equations Parts of an equation. In an equation: the variables are named unknowns (or indeterminate quantities), the number multiplied with a variable is named coefficient, a term is a summand of the equation, the highest power of the unknowns is called the order/degree of the equation. Example: In the equation 2x3 + 4y + 1 = 4: the unknowns are x and y, the coefficient of x3 is 2 and the coefficient of y is 4, the order of the equation is 3. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 15. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving Alberto Pardo Milan´s e Algebraic expressions and equations
  • 16. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving Solution of an equation. You are solving a equation when you replace a variable with a value and the mathematical expressions are still the same. The value for the variables is the solution of the equation. Example: In the equation 2x = 10 the solution is 5, because 2 · 5 = 10. Example: Sam is 9 years old. This is seven years younger than her sister Rose’s age. We can solve an equation to find Rose’s age: x − 7 = 9, the solution of the equation is 16, so Rose is 16 years old. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 17. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving Solution of an equation. You are solving a equation when you replace a variable with a value and the mathematical expressions are still the same. The value for the variables is the solution of the equation. Example: In the equation 2x = 10 the solution is 5, because 2 · 5 = 10. Example: Sam is 9 years old. This is seven years younger than her sister Rose’s age. We can solve an equation to find Rose’s age: x − 7 = 9, the solution of the equation is 16, so Rose is 16 years old. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 18. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving Solution of an equation. You are solving a equation when you replace a variable with a value and the mathematical expressions are still the same. The value for the variables is the solution of the equation. Example: In the equation 2x = 10 the solution is 5, because 2 · 5 = 10. Example: Sam is 9 years old. This is seven years younger than her sister Rose’s age. We can solve an equation to find Rose’s age: x − 7 = 9, the solution of the equation is 16, so Rose is 16 years old. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 19. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. To solve equations you can use the balance method, you must carry out the same operations in both sides and in the same order. You must use these properties: • Addition Property of Equalities: If you add the same number to each side of an equation, the two sides remain equal (note you can also add negative numbers). Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2 • Multiplication Property of Equalities: If you multiply by the same number each side of an equation, the two sides remain equal (note you can also multiply by fractions). x x Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30 5 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 20. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. To solve equations you can use the balance method, you must carry out the same operations in both sides and in the same order. You must use these properties: • Addition Property of Equalities: If you add the same number to each side of an equation, the two sides remain equal (note you can also add negative numbers). Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2 • Multiplication Property of Equalities: If you multiply by the same number each side of an equation, the two sides remain equal (note you can also multiply by fractions). x x Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30 5 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 21. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. To solve equations you can use the balance method, you must carry out the same operations in both sides and in the same order. You must use these properties: • Addition Property of Equalities: If you add the same number to each side of an equation, the two sides remain equal (note you can also add negative numbers). Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2 • Multiplication Property of Equalities: If you multiply by the same number each side of an equation, the two sides remain equal (note you can also multiply by fractions). x x Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30 5 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 22. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. To solve equations you can use the balance method, you must carry out the same operations in both sides and in the same order. You must use these properties: • Addition Property of Equalities: If you add the same number to each side of an equation, the two sides remain equal (note you can also add negative numbers). Example: x + 3 = 5 =⇒ x + 3−3 = 5−3 =⇒ x = 2 • Multiplication Property of Equalities: If you multiply by the same number each side of an equation, the two sides remain equal (note you can also multiply by fractions). x x Example: = 6 =⇒ 5 · = 5 · 6 =⇒ x = 30 5 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 23. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 24. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 25. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 26. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 27. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 28. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 29. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Solving The balance method. • Brackets: Sometimes you will need to solve equations involving brackets. If brackets appear, first remove the brackets by expanding each bracketed expression. Example: 2(x − 3) = 2 =⇒ 2x − 6 = 2 =⇒ 2x − 6+6 = 2+6 =⇒ 2x 8 2x = 8 =⇒ = =⇒ x = 4 2 2 Use all three properties to solve equations: Example: Solve 4x + 3 · (x − 25) = 240: First we remove brackets: 3 · (x − 25) = 3x − 75 so 4x+3x − 75 = 240. Them we use addition property: 4x+3x−75+75 = 240+75 =⇒ 4x+3x = 240+75 =⇒ 7x = 315. 7x 315 Now we can use multiplication property: = 7 7 So the solution is x = 45. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 30. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Alberto Pardo Milan´s e Algebraic expressions and equations
  • 31. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: a 4x + 2 = 26 b 5(2x − 1) = 7(9 − x) x 2x c + =7 2 3 d 19 + 4x = 9 − x e 3(2x + 1) = x − 2 x 3x 1 f − = 5 10 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 32. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: x 2x a 4x + 2 = 26 c + =7 2 3 4x = 26 − 2 3x 4x 42 + = 4x = 24 6 6 6 x = 24 : 4 3x + 4x = 42 x=6 7x = 42 x = 42 : 7 b 5(2x − 1) = 7(9 − x) x=6 10x − 5 = 63 − 7x 10x + 7x = 63 + 5 d 19 + 4x = 9 − x 17x = 68 4x + x = 9 − 19 x = 68 : 17 5x = −10 x=4 x = −10 : 5 x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 33. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: x 2x a 4x + 2 = 26 c + =7 2 3 4x = 26 − 2 3x 4x 42 + = 4x = 24 6 6 6 x = 24 : 4 3x + 4x = 42 x=6 7x = 42 x = 42 : 7 b 5(2x − 1) = 7(9 − x) x=6 10x − 5 = 63 − 7x 10x + 7x = 63 + 5 d 19 + 4x = 9 − x 17x = 68 4x + x = 9 − 19 x = 68 : 17 5x = −10 x=4 x = −10 : 5 x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 34. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: x 2x a 4x + 2 = 26 c + =7 2 3 4x = 26 − 2 3x 4x 42 + = 4x = 24 6 6 6 x = 24 : 4 3x + 4x = 42 x=6 7x = 42 x = 42 : 7 b 5(2x − 1) = 7(9 − x) x=6 10x − 5 = 63 − 7x 10x + 7x = 63 + 5 d 19 + 4x = 9 − x 17x = 68 4x + x = 9 − 19 x = 68 : 17 5x = −10 x=4 x = −10 : 5 x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 35. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: x 2x a 4x + 2 = 26 c + =7 2 3 4x = 26 − 2 3x 4x 42 + = 4x = 24 6 6 6 x = 24 : 4 3x + 4x = 42 x=6 7x = 42 x = 42 : 7 b 5(2x − 1) = 7(9 − x) x=6 10x − 5 = 63 − 7x 10x + 7x = 63 + 5 d 19 + 4x = 9 − x 17x = 68 4x + x = 9 − 19 x = 68 : 17 5x = −10 x=4 x = −10 : 5 x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 36. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: e 3(2x + 1) = x − 2 6x + 3 = x − 2 6x − x = −2 − 3 5x = −5 x = −5 : 5 x = −1 x 3x 1 f − = 5 10 5 2x 3x 2 − = 10 10 10 2x − 3x = 2 −1x = 2 x = 2 : (−1) x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 37. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 1 Solve the equations: e 3(2x + 1) = x − 2 6x + 3 = x − 2 6x − x = −2 − 3 5x = −5 x = −5 : 5 x = −1 x 3x 1 f − = 5 10 5 2x 3x 2 − = 10 10 10 2x − 3x = 2 −1x = 2 x = 2 : (−1) x = −2 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 38. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 2 Find a number such that 2 less than three times the number is 10. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 39. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 2 Find a number such that 2 less than three times the number is 10. Data: Let x be the number. Three times the number is 3x. 2 less than three times the number is (3x − 2) and this is 10. 3x − 2 = 10 3x = 12 x = 12 : 3 x=4 Answer: The number is x = 4. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 40. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 2 Find a number such that 2 less than three times the number is 10. Data: Let x be the number. Three times the number is 3x. 2 less than three times the number is (3x − 2) and this is 10. 3x − 2 = 10 3x = 12 x = 12 : 3 x=4 Answer: The number is x = 4. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 41. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 2 Find a number such that 2 less than three times the number is 10. Data: Let x be the number. Three times the number is 3x. 2 less than three times the number is (3x − 2) and this is 10. 3x − 2 = 10 3x = 12 x = 12 : 3 x=4 Answer: The number is x = 4. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 42. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 3 Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155 pounds less than twice her husband’s money. How many pounds does Mr. Roberts have? How many pounds does Mrs. Roberts have? Alberto Pardo Milan´s e Algebraic expressions and equations
  • 43. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 3 Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155 pounds less than twice her husband’s money. How many pounds does Mr. Roberts have? How many pounds does Mrs. Roberts have? Data: They have 370 pounds. Mr. Roberts has x pounds. Twice Mr. Roberts’ money is 2x pounds. Mrs. Roberts has (2x − 155) pounds. x+(2x−155) = 370 x + 2x − 155 = 370 3x = 525 x = 525 : 3 x = 175 Answer: Mr. Roberts has 175 370 − 175 = 195 pounds. Mrs. Roberts has 195 pounds. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 44. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 3 Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155 pounds less than twice her husband’s money. How many pounds does Mr. Roberts have? How many pounds does Mrs. Roberts have? Data: They have 370 pounds. Mr. Roberts has x pounds. Twice Mr. Roberts’ money is 2x pounds. Mrs. Roberts has (2x − 155) pounds. x+(2x−155) = 370 x + 2x − 155 = 370 3x = 525 x = 525 : 3 x = 175 Answer: Mr. Roberts has 175 370 − 175 = 195 pounds. Mrs. Roberts has 195 pounds. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 45. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 3 Mr. Roberts and his wife have 370 pounds. Mrs. Roberts has 155 pounds less than twice her husband’s money. How many pounds does Mr. Roberts have? How many pounds does Mrs. Roberts have? Data: They have 370 pounds. Mr. Roberts has x pounds. Twice Mr. Roberts’ money is 2x pounds. Mrs. Roberts has (2x − 155) pounds. x+(2x−155) = 370 x + 2x − 155 = 370 3x = 525 x = 525 : 3 x = 175 Answer: Mr. Roberts has 175 370 − 175 = 195 pounds. Mrs. Roberts has 195 pounds. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 46. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 4 The length of a room exceeds the width by 5 feet. The length of the four walls is 30 feet. Find the dimensions of the room. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 47. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 4 The length of a room exceeds the width by 5 feet. The length of the four walls is 30 feet. Find the dimensions of the room. Data: The width of the room is x feet. The length of the room is (x + 5) feet. The length of the four walls is x + (x + 5) + x + (x + 5) feet and this is 30 feet. x + (x + 5) + x + (x + 5) = 30 x + x + 5 + x + x + 5 = 30 x + x + x + x = 30 − 5 − 5 4x = 20 x = 20 : 4 x=5 5 + 5 = 10 Answer: The room is 5 feet long and 10 feet wide. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 48. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 4 The length of a room exceeds the width by 5 feet. The length of the four walls is 30 feet. Find the dimensions of the room. Data: The width of the room is x feet. The length of the room is (x + 5) feet. The length of the four walls is x + (x + 5) + x + (x + 5) feet and this is 30 feet. x + (x + 5) + x + (x + 5) = 30 x + x + 5 + x + x + 5 = 30 x + x + x + x = 30 − 5 − 5 4x = 20 x = 20 : 4 x=5 5 + 5 = 10 Answer: The room is 5 feet long and 10 feet wide. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 49. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 4 The length of a room exceeds the width by 5 feet. The length of the four walls is 30 feet. Find the dimensions of the room. Data: The width of the room is x feet. The length of the room is (x + 5) feet. The length of the four walls is x + (x + 5) + x + (x + 5) feet and this is 30 feet. x + (x + 5) + x + (x + 5) = 30 x + x + 5 + x + x + 5 = 30 x + x + x + x = 30 − 5 − 5 4x = 20 x = 20 : 4 x=5 5 + 5 = 10 Answer: The room is 5 feet long and 10 feet wide. Alberto Pardo Milan´s e Algebraic expressions and equations
  • 50. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 5 Maria spent a third of her money on food. Then, she spent e21 on a present. At the end, she had the fifth of her money. How much money did she have at the beginning? Alberto Pardo Milan´s e Algebraic expressions and equations
  • 51. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 5 Maria spent a third of her money on food. Then, she spent e21 on a present. At the end, she had the fifth of her money. How much money did she have at the beginning? x Data: At the beginig She had x euros. She spent on food 3 x and e21 on a present. At the end She had euros. 5 x x x − − 21 = 7x = 315 3 5 x x 21 x x = 315 : 7 − − = x = 45 1 3 1 5 15x 5x 315 3x − − = 15 15 15 15 Answer: At the beginig 15x − 5x − 315 = 3x She had e45. 15x − 5x − 3x = 315 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 52. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 5 Maria spent a third of her money on food. Then, she spent e21 on a present. At the end, she had the fifth of her money. How much money did she have at the beginning? x Data: At the beginig She had x euros. She spent on food 3 x and e21 on a present. At the end She had euros. 5 x x x − − 21 = 7x = 315 3 5 x x 21 x x = 315 : 7 − − = x = 45 1 3 1 5 15x 5x 315 3x − − = 15 15 15 15 Answer: At the beginig 15x − 5x − 315 = 3x She had e45. 15x − 5x − 3x = 315 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 53. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 5 Maria spent a third of her money on food. Then, she spent e21 on a present. At the end, she had the fifth of her money. How much money did she have at the beginning? x Data: At the beginig She had x euros. She spent on food 3 x and e21 on a present. At the end She had euros. 5 x x x − − 21 = 7x = 315 3 5 x x 21 x x = 315 : 7 − − = x = 45 1 3 1 5 15x 5x 315 3x − − = 15 15 15 15 Answer: At the beginig 15x − 5x − 315 = 3x She had e45. 15x − 5x − 3x = 315 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 54. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 6 John bought a book, a pencil and a notebook. The book cost the double of the notebook, and the pencil cost the fifth of the book and the notebook together. If he paid e18, what is the price of each article? Alberto Pardo Milan´s e Algebraic expressions and equations
  • 55. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 6 John bought a book, a pencil and a notebook. The book cost the double of the notebook, and the pencil cost the fifth of the book and the notebook together. If he paid e18, what is the price of each article? Data: The notebook cost x euros. The book cost 2x euros. The notebook and the book together cost (x + 2x) euros. x + 2x The pencil cost . He paid e18. 5 x + 2x 5x+10x+x+2x = 90 x + 2x + = 18 5 18x = 90 x 2x x + 2x 18 x = 90 : 18 + + = 1 1 5 1 x=5 5x 10x x + 2x 90 2x = 10 + + = x + 2x 15 5 5 5 5 = = 3. 5x + 10x + (x + 2x) = 90 5 5 Alberto Pardo Milan´s e Algebraic expressions and equations
  • 56. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 6 Data: The notebook cost x euros. The book cost 2x euros. The notebook and the book together cost (x + 2x) euros. x + 2x The pencil cost . He paid e18. 5 x + 2x 5x+10x+x+2x = 90 x + 2x + = 18 5 18x = 90 x 2x x + 2x 18 x = 90 : 18 + + = 1 1 5 1 x=5 5x 10x x + 2x 90 2x = 10 + + = x + 2x 15 5 5 5 5 = = 3. 5x + 10x + (x + 2x) = 90 5 5 Answer: The notebook cost e5, the book e10 and the pencil e3 . Alberto Pardo Milan´s e Algebraic expressions and equations
  • 57. Indice Monomials Adding and subtracting Identities and Equations Solving Exercises Exercises Exercise 6 Data: The notebook cost x euros. The book cost 2x euros. The notebook and the book together cost (x + 2x) euros. x + 2x The pencil cost . He paid e18. 5 x + 2x 5x+10x+x+2x = 90 x + 2x + = 18 5 18x = 90 x 2x x + 2x 18 x = 90 : 18 + + = 1 1 5 1 x=5 5x 10x x + 2x 90 2x = 10 + + = x + 2x 15 5 5 5 5 = = 3. 5x + 10x + (x + 2x) = 90 5 5 Answer: The notebook cost e5, the book e10 and the pencil e3 . Alberto Pardo Milan´s e Algebraic expressions and equations