2.3
Combining the Rules to Solve
Equations
2.3
OBJECTIVES
1. Combine the addition and multiplication
properties to solve an equation
2. Use the order of operations when solving
an equation
3. Recognize identities
4. Recognize equations with no solutions
In all our examples thus far, either the addition property or the multiplication property was
used in solving an equation. Often, finding a solution will require the use of both
properties.
Example 1
Solving Equations
(a) Solve
4x 5 7
Here x is multiplied by 4. The result, 4x, then has 5 subtracted from it (or 5 added to it)
on the left side of the equation. These two operations mean that both properties must be
applied in solving the equation.
Because the variable term is already on the left, we start by adding 5 to both sides:
4x 5 7
5 5
4x
12
We now divide both sides by 4:
12
4x
4
4
x3
The solution is 3. To check, replace x with 3 in the original equation. Be careful to follow
the rules for the order of operations.
4357
12 5 7
77
(True)
© 2001 McGraw-Hill Companies
(b) Solve
3x 8 4
8 8
3x
Add 8 to both sides.
12
Now divide both sides by 3 to isolate x on the left.
12
3x
3
3
x 4
The solution is 4. We’ll leave the check of this result to you.
165
166
CHAPTER 2
EQUATIONS AND INEQUALITIES
CHECK YOURSELF 1
Solve and check.
(a) 6x 9 15
(b) 5x 8 7
The variable may appear in any position in an equation. Just apply the rules carefully as
you try to write an equivalent equation, and you will find the solution. Example 2 illustrates
this property.
Example 2
Solving Equations
Solve
3 2x 9
3
3
First add 3 to both sides.
2x 6
Now divide both sides by 2. This will leave x alone on the left.
6
2x
2
2
x 3
The solution is 3. We’ll leave it to you to check this result.
CHECK YOURSELF 2
Solve and check.
10 3x 1
You may also have to combine multiplication with addition or subtraction to solve an
equation. Consider Example 3.
Example 3
Solving Equations
(a) Solve
x
34
5
To get the x term alone, we first add 3 to both sides.
x
3 4
5
3 3
x
7
5
Now, to undo the division multiply both sides of the equation by 5.
5
5 5 7
x
x 35
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2
NOTE
1, so we divide by
2
2 to isolate x on the left.
COMBINING THE RULES TO SOLVE EQUATIONS
SECTION 2.3
167
The solution is 35. Just return to the original equation to check the result.
35
3 4
5
734
4 4
(True)
(b) Solve
2
x 5 13
3
5 5
2
x
3
First add 5 to both sides.
8
2
3
Now multiply both sides by , the reciprocal of .
2
3
3 2
3
x 8
2 3
2
or
x 12
The solution is 12. We’ll leave it to you to check this result.
CHECK YOURSELF 3
Solve and check.
(a)
x
53
6
(b)
3
x 8 10
4
In Section 2.1, you learned how to solve certain equations when the variable appeared
on both sides. Example 4 will show you how to extend that work by using the multiplication property of equality.
Example 4
Solving an Equation
Solve
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6x 4 3x 2
First add 4 to both sides. This will undo the subtraction on the left.
6x 4 3x 2
4
4
6x
3x 2
Now add 3x so that the terms in x will be on the left only.
6x 3x 2
3x 3x
3x 2
CHAPTER 2
EQUATIONS AND INEQUALITIES
Finally divide by 3.
3x
2
3
3
x
2
3
Check:
6
3 4 33 2
2
2
4422
00
(True)
As you know, the basic idea is to use our two properties to form an equivalent equation
with the x isolated. Here we added 4 and then subtracted 3x. You can do these steps in either
order. Try it for yourself the other way. In either case, the multiplication property is then
used as the last step in finding the solution.
CHECK YOURSELF 4
Solve and check.
7x 5 3x 5
Let’s look at two approaches to solving equations in which the coefficient on the right
side is greater than the coefficient on the left side.
Example 5
Solving an Equation (Two Methods)
Solve 4x 8 7x 7.
Method 1
4x 8 8
4x
7x
3x
7x 7
8
7x 15
7x
3x
15
3
3
Adding 8 will leave the x term alone
on the left.
Adding 7x will get the variable
terms on the left.
15
Dividing by 3 will isolate x on the left.
x 5
We’ll let you check this result.
To avoid a negative coefficient (in Example 5, 3), some students prefer a different
approach.
This time we’ll work toward having the number on the left and the x term on the right, or
x
© 2001 McGraw-Hill Companies
168
COMBINING THE RULES TO SOLVE EQUATIONS
SECTION 2.3
169
Method 2
NOTE It is usually easier to
isolate the variable term on the
side that will result in a positive
coefficient.
4x 8 7x 7
7
7
4x 15 7x
4x
4x
15 3x
3x
15
3
3
Add 7.
Add 4x to get the variables
on the right.
Dividing by 3 to isolate x on the right.
5 x
Because 5 x and x 5 are equivalent equations, it really makes no difference; the
solution is still 5! You can use whichever approach you prefer.
CHECK YOURSELF 5
Solve 5x 3 9x 21 by finding equivalent equations of the form x x to compare the two methods of finding the solution.
and
It may also be necessary to remove grouping symbols in solving an equation.
Example 6
Solving Equations That Contain Parentheses
Solve and check.
NOTE
5(x 3)
5 (x (3))
5x 5 (3)
5x (15)
5x 15
5(x 3) 2x x 7
5x 15 2x x 7
3x 15 x 7
15
15
3x x 22
x x
2x 22
First, apply the distributive property.
Combine like terms.
Add 15.
Add x.
Divide by 2.
x 11
The solution is 11. To check, substitute 11 for x in the original equation. Again note the use
of our rules for the order of operations.
5(11 3) 2 11 11 7
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5 8 2 11 11 7
Simplify terms in parentheses.
Multiply.
40 22 11 7
Add and subtract.
18 18
A true statement.
CHECK YOURSELF 6
Solve and check.
7(x 5) 3x x 7
An equation that is true for any value of x is called an identity.
170
CHAPTER 2
EQUATIONS AND INEQUALITIES
Example 7
Solving an Equation
Solve the equation 2(x 3) 2x 6
2(x 3) 2x 6
2x 6 2x 6
2x
2x
NOTE We could ask the
question “For what values of x
does 6 6?”
6 6
The statement 6 6 is true for any value of x. The original equation is an identity.
CHECK YOURSELF 7
Solve the equation 3(x 4) 2x x 12
There are also equations for which there are no solutions.
Example 8
Solving an Equation
Solve the equation 3(2x 5) 4x 2x 1
3(2x 5) 4x 2x 1
6x 15 4x 2x 1
2x 15 2x 1
2x
2x
15 1
NOTE We could ask the
question “For what values of x
does 15 1?”
These two numbers are never equal. The original equation has no solutions.
CHECK YOURSELF 8
Solve the equation 2(x 5) x 3x 3
Step by Step:
is sometimes called an
algorithm for the process.
Step 1
Step 2
Step 3
Step 4
Solving Linear Equations
Use the distributive property to remove any grouping symbols. Then
simplify by combining like terms on each side of the equation.
Add or subtract the same term on each side of the equation until the
variable term is on one side and a number is on the other.
Multiply or divide both sides of the equation by the same nonzero
number so that the variable is alone on one side of the equation. If no
variable remains, determine whether the original equation is an
identity or whether it has no solutions.
Check the solution in the original equation.
CHECK YOURSELF ANSWERS
1. (a) 4; (b) 3
2. 3
3. (a) 12; (b) 24
7. The equation is an identity, x is any real number.
4.
5
5. 6
6. 14
2
8. There are no solutions.
© 2001 McGraw-Hill Companies
NOTE Such an outline of steps
Name
2.3
Exercises
Section
Date
Solve for x and check your result.
1. 2x 1 9
2. 3x 1 17
ANSWERS
1.
3. 3x 2 7
4. 5x 3 23
2.
3.
5. 4x 7 35
6. 7x 8 13
4.
5.
7. 2x 9 5
8. 6x 25 5
6.
7.
9. 4 7x 18
10. 8 5x 7
8.
9.
10.
11. 3 4x 9
12. 5 4x 25
11.
12.
13.
x
15
2
14.
x
23
3
13.
14.
15.
x
53
4
16.
x
38
5
15.
16.
2
17. x 5 17
3
3
18. x 5 4
4
17.
18.
19.
© 2001 McGraw-Hill Companies
19.
4
x 3 13
5
20.
5
x 4 14
7
20.
21.
21. 5x 2x 9
22. 7x 18 2x
22.
23.
23. 3x 10 2x
24. 11x 7x 20
24.
171
ANSWERS
25.
25. 9x 2 3x 38
26. 8x 3 4x 17
27. 4x 8 x 14
28. 6x 5 3x 29
29. 5x 7 2x 3
30. 9x 7 5x 3
31. 7x 3 9x 5
32. 5x 2 8x 11
33. 5x 4 7x 8
34. 2x 23 6x 5
35. 2x 3 5x 7 4x 2
36. 8x 7 2x 2 4x 5
37. 6x 7 4x 8 7x 26
38. 7x 2 3x 5 8x 13
39. 9x 2 7x 13 10x 13
40. 5x 3 6x 11 8x 25
41. 8x 7 5x 10 10x 12
42. 10x 9 2x 3 8x 18
43. 7(2x 1) 5x x 25
44. 9(3x 2) 10x 12x 7
45. 3x 2(4x 3) 6x 9
46. 7x 3(2x 5) 10x 17
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
46.
47.
48.
47.
8
2
x 3 x 15
3
3
48.
12
3
x 7 31 x
5
5
49.
2
12
x5
x8
5
5
50.
3
24
x5
x7
7
7
49.
50.
172
© 2001 McGraw-Hill Companies
45.
ANSWERS
51. 5.3x 7 2.3x 5
52. 9.8x 2 3.8x 20
51.
52.
53.
53. 4(x 5) 4x 20
54. 3(2x 4) 12 6x
54.
55.
55. 5(x 1) 4x x 5
56. 4(2x 3) 8x 5
56.
57.
57. 6x 4x 1 12 2x 11
58. 2x 5x 9 3(x 4) 5
58.
59.
59. 4(x 2) 11 2(2x 3) 13
60. 4(x 2) 5 2(2x 7)
60.
61.
61. Create an equation of the form ax b c that has 2 as a solution.
62.
62. Create an equation of the form ax b c that has 7 as a solution.
63.
63. The equation 3x 3x 5 has no solution, whereas the equation 7x 8 8 has
64.
zero as a solution. Explain the difference between a solution of zero and no
solution.
65.
66.
64. Construct an equation for which every real number is a solution.
67.
68.
In exercises 65 to 68, find the length of each side of the figure for the given perimeter.
65.
2x 2
x
66.
3x 4
x
P 32 cm
P 24 in.
67.
68.
4x 5
1
3x 2
3x 3x
2
2x
P 34 cm
1
x
© 2001 McGraw-Hill Companies
x2
P 90 in.
173
ANSWERS
a.
Getting Ready for Section 2.4 [Section 1.7]
b.
Divide.
c.
3b
3
4xy
(c)
4x
7mn2
(e)
7n2
srt
(g)
sr
d.
5x
5
6a2b
(d)
6a2
pab
(f)
pa
x2yz
(h) 2
xz
(a)
e.
f.
g.
h.
(b)
Answers
1. 4
15. 32
29. 43. 4
3. 3
5. 7
7. 2
9. 2
11. 3
13. 8
17. 18
19. 20
21. 3
23. 2
25. 6
27. 2
10
3
31. 4
45. 55. No solution
63.
b. x
35. 4
37. 5
39. 4
41.
13
51. 4
53. Identity
2
57. Identity
59. Identity
61. 6x 5 17
65. 6 in., 8 in., 10 in.
67. 12 in., 19 in., 29 in., 30 in.
49. 47. 9
c. y
5
3
d. b
e. m
f. b
g. t
h. y
© 2001 McGraw-Hill Companies
a. b
3
5
33. 6
174