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              Today:
             Warm-Up
     Review Systems of Equations
       New Solving Techniques
Warm Up

1.
Warm Up


2. Solve for a: 9a – 2b = c + 4a       3. Solve by graphing and
                                            state the solution
4. a. Write the equation of the line
   b. Write the inequality of the line.
Warm Up

5. If it takes eight carpenters (all working at the same
   time and rate) fifteen days to build a house, how long
   will it take for ten carpenters to build the same house?

  8 carpenters • 15 days = 120 carpenter-days.
  Therefore 10x (x being the number of days)= 120.
  x = 12 days for 10 carpenters
5. 1/3(3/5x + 18) = 3/4x +2/5
Review: Solve by Elimination

 Practice 1: y = x + 7
             x + 2y = 5

Practice 2: x - 3y = 7
            3x +3y = 9
Solve: Elimination By Multiplying


                          0x + y = = 4
                          x + 0y 4
 Like variables
 must be lined            2x + 3y = 9
 under each other.




          We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If there was
a –2x in the 1st equation, the x’s would be eliminated when we
add. So we will multiply the 1st equation by a – 2.
Solve: Elimination By Multiplying

( X + Y = 4 ) -2               -2X - 2 Y = - 8
2X + 3Y = 9                    2X + 3Y = 9

   Now add the two equations        Y=1
   and solve.




             THEN----
Substitute your answer into either original equation and
solve for the second variable.

                    X+Y=4
                   X +1=4
                     - 1 -1
                     X=3
                 Answer               (3,1)
  Now check our answers in both equations--
Solve: Elimination By Multiplying

       x+y=4
       3+1=4
         4=4


     2x + 3y = 9
  2(3) + 3(1) = 9
        6+3=9
            9=9
Solve: By Substitution
 Recall that when we 'solve' a point-slope formula, we
 end up in slope-intercept form. In much the same
 way, the substitution method is closely related to the
 elimination method.
After eliminating one variable and solving for the other, we
substitute the value of the variable back into the equation.
For example: Solve 2x + 3y = -26 using elimination
                           4x - 3y = 2

  What is the      At this point we substitute -4 for x,
  value of x ?     and solve for y. This is exactly
                   what the substitution method is
                   except it is done at the beginning.
Solve: By Substitution
  Example 1: y = 2x
              4x - y = -
  4
Example 1: Substitute 2x for y in the 2nd equation
             y = 2x
             4x - 2x = -4; 2x = -4; x = -2

Then, substitute -2 for x in the first equation:
                  y = 2(-2); y = -4
Finally, plug both values in and check for equality.

 -4 = 2(-2); True; 4(-2) - (-4) = -4; -8 + 4 = -4; True
Solve: By Substitution

Example 2: 3x + 5y = -7
           x = 2y + 5


Example 3: y = 2x - 1
           6x - 3y = 7
Applying Systems of Equations
            Solve by Elimination.
Example 1: The sum of two numbers is 52.
The larger number is 2 more than 4 times
the smaller number. Find both numbers.
Example 1:   -(x + y = -52
              -x - y = 52)
                  x+y=
52         + _________ Rearrange
               x = 4y + 2
                 -4y = 2
                  -5y = -50 y = 10

           x + 10 = 52; x = 42
Class Work:
January 23
January 23

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January 23

  • 1. 23 Today: Warm-Up Review Systems of Equations New Solving Techniques
  • 3. Warm Up 2. Solve for a: 9a – 2b = c + 4a 3. Solve by graphing and state the solution 4. a. Write the equation of the line b. Write the inequality of the line.
  • 4. Warm Up 5. If it takes eight carpenters (all working at the same time and rate) fifteen days to build a house, how long will it take for ten carpenters to build the same house? 8 carpenters • 15 days = 120 carpenter-days. Therefore 10x (x being the number of days)= 120. x = 12 days for 10 carpenters 5. 1/3(3/5x + 18) = 3/4x +2/5
  • 5. Review: Solve by Elimination Practice 1: y = x + 7 x + 2y = 5 Practice 2: x - 3y = 7 3x +3y = 9
  • 6. Solve: Elimination By Multiplying 0x + y = = 4 x + 0y 4 Like variables must be lined 2x + 3y = 9 under each other. We need to eliminate (get rid of) a variable. To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.
  • 7. Solve: Elimination By Multiplying ( X + Y = 4 ) -2 -2X - 2 Y = - 8 2X + 3Y = 9 2X + 3Y = 9 Now add the two equations Y=1 and solve. THEN----
  • 8. Substitute your answer into either original equation and solve for the second variable. X+Y=4 X +1=4 - 1 -1 X=3 Answer (3,1) Now check our answers in both equations--
  • 9. Solve: Elimination By Multiplying x+y=4 3+1=4 4=4 2x + 3y = 9 2(3) + 3(1) = 9 6+3=9 9=9
  • 10. Solve: By Substitution Recall that when we 'solve' a point-slope formula, we end up in slope-intercept form. In much the same way, the substitution method is closely related to the elimination method. After eliminating one variable and solving for the other, we substitute the value of the variable back into the equation. For example: Solve 2x + 3y = -26 using elimination 4x - 3y = 2 What is the At this point we substitute -4 for x, value of x ? and solve for y. This is exactly what the substitution method is except it is done at the beginning.
  • 11. Solve: By Substitution Example 1: y = 2x 4x - y = - 4 Example 1: Substitute 2x for y in the 2nd equation y = 2x 4x - 2x = -4; 2x = -4; x = -2 Then, substitute -2 for x in the first equation: y = 2(-2); y = -4 Finally, plug both values in and check for equality. -4 = 2(-2); True; 4(-2) - (-4) = -4; -8 + 4 = -4; True
  • 12. Solve: By Substitution Example 2: 3x + 5y = -7 x = 2y + 5 Example 3: y = 2x - 1 6x - 3y = 7
  • 13. Applying Systems of Equations Solve by Elimination. Example 1: The sum of two numbers is 52. The larger number is 2 more than 4 times the smaller number. Find both numbers. Example 1: -(x + y = -52 -x - y = 52) x+y= 52 + _________ Rearrange x = 4y + 2 -4y = 2 -5y = -50 y = 10 x + 10 = 52; x = 42