1. 1.3
Using the Principles Together
OBJECTIVES
a Solve equations using both the addition principle and
the multiplication principle.
b Solve equations in which like terms may need to be
collected.
c Solve equations by first removing parentheses and
collecting like terms; solve equations with an infinite
number of solutions and equations with no solutions.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 1

2. 1.3
Using the Principles Together
Solve equations using both the addition principle and
a
the multiplication principle.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2

3. 1.3
Using the Principles Together
a
EXAMPLE
Solve equations using both the addition principle and
the multiplication principle.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3

4. 1.3
Using the Principles Together
Solve equations using both the addition principle and
a
the multiplication principle.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4

5. 1.3
Using the Principles Together
b
Solve equations in which like terms may need to be
collected.
If there are like terms on one side of the equation, we
collect them before using the addition principle or the
multiplication principle.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 5

6. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6

7. 1.3
Using the Principles Together
b Solve equations in which like terms may need to be
collected.
If there are like terms on opposite sides of the equation,
we get them on the same side by using the addition
principle. Then we collect them. In other words, we get
all the terms with a variable on one side of the equation
and all the terms without a variable on the other side.
If there are like terms on one side at the outset, they
should be collected first.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7

8. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 8

9. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 9

10. 1.3
b
Using the Principles Together
Solve equations in which like terms may need to be
collected.
In general, equations are easier to solve if they do not
contain fractions or decimals.
The easiest way to clear an equation of fractions is to
multiply every term on both sides by the least common
multiple of all the denominators.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 10

11. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
The denominators are 3, 6, and 2. The number 6 is the
least common multiple of all the denominators. We
multiply by 6 on both sides of the equation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11

12. 1.3
b
Using the Principles Together
Solve equations in which like terms may need to be
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12

13. 1.3
b
Using the Principles Together
Solve equations in which like terms may need to be
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 13

14. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 14

15. 1.3
Using the Principles Together
b
Solve equations in which like terms may need to be
collected.
To clear an equation of decimals, we count the greatest
number of decimal places in any one number. If the
greatest number of decimal places is 1, we multiply
every term on both sides by 10; if it is 2, we multiply by
100; and so on.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 15

16. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
The greatest number of decimal places in any one number
is two. Multiplying by 100, which has two 0’s, will clear all
decimals.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 16

17. 1.3
Using the Principles Together
Solve equations in which like terms may need to be
b
collected.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 17

18. 1.3
Using the Principles Together
c
Solve equations by first removing parentheses and
collecting like terms; solve equations with an infinite
number of solutions and equations with no solutions.
To solve certain kinds of equations that contain
parentheses, we first use the distributive laws to remove
the parentheses. Then we proceed as before.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 18

19. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting
like terms; solve equations with an infinite number of solutions
and equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 19

20. 1.3
Using the Principles Together
1. Multiply on both sides to clear the equation of
fractions or decimals. (This is optional, but it can ease
computations.)
2. If parentheses occur, multiply to remove them using
the distributive laws.
3. Collect like terms on each side, if necessary.
4. Get all terms with variables on one side and all
numbers (constant terms) on the other side, using the
addition principle.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 20

21. 1.3
Using the Principles Together
5. Collect like terms again, if necessary.
6. Multiply or divide to solve for the variable, using the
multiplication principle.
7. Check all possible solutions in the original equation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 21

22. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting like
terms; solve equations with an infinite number of solutions and
equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 22

23. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting like
terms; solve equations with an infinite number of solutions and
equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 23

24. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting like
terms; solve equations with an infinite number of solutions and
equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 24

25. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting like
terms; solve equations with an infinite number of solutions and
equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 25

26. 1.3
Using the Principles Together
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 26

27. 1.3
c
Using the Principles Together
Solve equations by first removing parentheses and collecting like
terms; solve equations with an infinite number of solutions and
equations with no solutions.
EXAMPLE
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 27