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Combining the Properties to Solve Equations
SECTION 11.6
755
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.
c
Example 2
Solving Equations
Solve
3 Ϫ 2x ϭ 9
3 Ϫ 3 Ϫ 2x ϭ 9 Ϫ 3
Ϫ2x ϭ 6
First subtract 3 from both sides.
Now divide both sides by Ϫ2. This leaves x alone on the left.
NOTE
Ϫ2
ϭ 1, so we divide by Ϫ2
Ϫ2
to isolate x.
Ϫ2x
6
ϭ
Ϫ2
Ϫ2
x ϭ Ϫ3
The solution is Ϫ3. We leave it to you to check this result.
© The McGrawHill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
Check Yourself 2
Solve and check.
10 ؊ 3x ؍1
Apago PDF Enhancer
You may also have to combine multiplication with addition or subtraction to solve
an equation. Consider Example 3.
c
Example 3
Solving Equations
(a) Solve
x
Ϫ3ϭ4
5
To get the x term alone, we first add 3 to both sides.
x
Ϫ3ϩ3ϭ4ϩ3
5
x
ϭ7
5
To undo the division, multiply both sides of the equation by 5.
5
5 ϭ 5 # 7
x
x ϭ 35
The solution is 35. Just return to the original equation to check the result.
(35)
Ϫ34
5
7Ϫ34
4ϭ4
(True)
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CHAPTER 11
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An Introduction to Algebra
(b) Solve
2
x ϩ 5 ϭ 13
3
2
x ϩ 5 Ϫ 5 ϭ 13 Ϫ 5
3
2
xϭ8
3
First subtract 5 from both sides.
3
2
Now multiply both sides by , the reciprocal of .
2
3
3 2
3
x ϭ
8
2 3
2
or
x ϭ 12
The solution is 12. We leave it to you to check this result.
x
؉5؍3
6
(b)
3
x ؊ 8 ؍10
4
Apago PDF Enhancer
In Section 11.4, you learned how to solve certain equations when the variable
appeared on both sides. Example 4 shows you how to extend that work by using the
multiplication property of equality.
c
Example 4
Combining Properties to Solve an Equation
Solve
6x Ϫ 4 ϭ 3x Ϫ 2
We begin by bringing all the variable terms to one side. To do this, we subtract 3x
from both sides. This eliminates the variable term from the right side.
6x Ϫ 4 ϭ 3x Ϫ 2
6x Ϫ 4 Ϫ 3x ϭ 3x Ϫ 2 Ϫ 3x
3x Ϫ 4 ϭ Ϫ2
We now isolate the variable term by adding 4 to both sides.
3x Ϫ 4 ϭ Ϫ2
3x Ϫ 4 ϩ 4 ϭ Ϫ2 ϩ 4
3x ϭ 2
Finally, divide by 3.
3x
2
ϭ
3
3
2
xϭ
3
The Streeter/Hutchison Series in Mathematics
(a)
© The McGrawHill Companies. All Rights Reserved.
Solve and check.
Basic Mathematical Skills with Geometry
Check Yourself 3
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Combining the Properties to Solve Equations
SECTION 11.6
757
Check:
6
3 Ϫ 4 33 Ϫ 2
2
2
4 Ϫ 42Ϫ2
0ϭ0
(True)
The basic idea is to use our two properties to form an equivalent equation with the
x isolated. Here we subtracted 3x and then added 4. 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
c
Example 5
© The McGrawHill Companies. All Rights Reserved.
Combining Properties to Solve an Equation (Two Methods)
Solve 4x Ϫ 8 ϭ 7x ϩ 7.
Method 1
Apago PDF Enhancer
4x Ϫ 8 Ϫ 7x ϭ 7x ϩ 7 Ϫ 7x
Ϫ3x Ϫ 8 ϭ 7
Ϫ3x Ϫ 8 ϩ 8 ϭ 7 ϩ 8
Ϫ3x ϭ 15
Ϫ3x
15
ϭ
Ϫ3
Ϫ3
x ϭ Ϫ5
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
Here are two approaches to solving equations in which the coefficient on the right
side is greater than the coefficient on the left side.
Bring the variable terms to the same
(left) side.
Isolate the variable term.
Isolate the variable.
We let you check this result.
To avoid a negative coefficient (Ϫ3, in this example), some students prefer a different approach.
This time we work toward having the number on the left and the x term on the right, or
ϭ x.
Method 2
NOTE
It is usually easier to isolate
the variable term on the side
that results in a positive
coefficient.
4x Ϫ 8 ϭ 7x ϩ 7
4x Ϫ 8 Ϫ 4x ϭ 7x ϩ 7 Ϫ 4x
Ϫ8 ϭ 3x ϩ 7
Ϫ8 Ϫ 7 ϭ 3x ϩ 7 Ϫ 7
Ϫ15 ϭ 3x
Ϫ15
3x
ϭ
3
3
Ϫ5 ϭ x
Bring the variable terms to the same (right) side.
Isolate the variable term.
Isolate the variable.
Because Ϫ5 ϭ x and x ϭ Ϫ5 are equivalent equations, it really makes no difference;
the solution is still Ϫ5! You may use whichever approach you prefer.
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An Introduction to Algebra
Check Yourself 5
Solve 5x ؉ 3 ؍9x ؊ 21 by finding equivalent equations of the form
x؍
and
؍x to compare the two methods of finding the
solution.
When possible, we start by combining like terms on each side of the equation.
Solve.
7x Ϫ 3 ϩ 5x ϩ 4
12x ϩ 1
12x ϩ 1 Ϫ 6x
6x ϩ 1
6x ϩ 1 Ϫ 1
6x
6x
6
x
ϭ
ϭ
ϭ
ϭ
ϭ
ϭ
6x ϩ 25
6x ϩ 25
6x ϩ 25 Ϫ 6x
25
25 Ϫ 1
24
24
ϭ
6
ϭ4
Start by combining like terms.
Bring the variables to one side.
Isolate the variable term.
Isolate the variable.
The solution is 4. We leave the check to you.
Check Yourself 6
Apago PDF Enhancer
Solve and check.
9x ؊ 6 ؊ 3x ؉ 1 ؍2x ؉ 15
It may also be necessary to remove grouping symbols to solve an equation.
Example 7 illustrates this property.
c
Example 7
Solving Equations That Contain Parentheses
Solve and check.
NOTE
5(x Ϫ 3)
ϭ 5[x ϩ (Ϫ3)]
ϭ 5x ϩ 5(Ϫ3)
ϭ 5x ϩ (Ϫ15)
ϭ 5x Ϫ 15
5(x Ϫ 3) Ϫ 2x ϭ x ϩ 7
5x Ϫ 15 Ϫ 2x ϭ x ϩ 7
3x Ϫ 15 ϭ x ϩ 7
Apply the distributive property.
Combine like terms.
We now have an equation that we can solve by the usual methods. First, bring the variable terms to one side, then isolate the variable term, and finally, isolate the variable.
3x Ϫ 15 Ϫ x
2x Ϫ 15
2x Ϫ 15 ϩ 15
2x
2x
2
x
ϭ
ϭ
ϭ
ϭ
xϩ7Ϫx
7
7 ϩ 15
22
22
ϭ
2
ϭ 11
Subtract x to bring the variable
terms to the same side.
Add 15 to isolate the variable term.
Divide by 2 to isolate the variable.
Basic Mathematical Skills with Geometry
< Objective 2 >
Combining Terms to Solve an Equation
The Streeter/Hutchison Series in Mathematics
Example 6
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c
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Combining the Properties to Solve Equations
SECTION 11.6
759
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) Ϫ 3] Ϫ 2(11) (11) ϩ 7
5 ؒ 8 Ϫ 2 ؒ 11 11 ϩ 7
40 Ϫ 22 11 ϩ 7
18 ϭ 18
Simplify terms in parentheses.
Multiply.
Add and subtract.
A true statement.
Check Yourself 7
Solve and check.
7(x ؉ 5) ؊ 3x ؍x ؊ 7
We say that an equation is “solved” when we have an equivalent equation of the form
xϭ
or
ϭx
in which
is some number
The steps of solving a linear equation are as follows:
© The McGrawHill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
Step by Step
To Solve a Linear
Equation
Step 1
Use the distributive property to remove any grouping symbols.
Step 2
Combine like terms on each side of the equation.
Step 3
Add or subtract variable terms to bring the variable term to one side
of the equation.
Step 4
Add or subtract numbers to isolate the variable term.
Step 5
Multiply by the reciprocal of the coefficient to isolate the variable.
Step 6
Check your result.
Apago PDF Enhancer
There are a host of applications involving linear equations.
c
Example 8
Applying Algebra
In an election, the winning candidate had 160 more votes than the loser did. If the total
number of votes cast was 3,260, how many votes did each candidate receive?
We first set up the problem. Let x represent the number of votes received by the
loser. Then the winner received x ϩ 160 votes.
We can set up an equation by adding the number of votes the candidates received.
This must total 3,260.
x ϩ (x ϩ 160) ϭ 3,260
2x ϩ 160 ϭ 3,260
2x ϭ 3,100
x ϭ 1,550
Remove the parentheses and combine like terms.
Subtract 160 from both sides.
Divide both sides by 2.
The loser received 1,550 votes. Therefore, the winner received
x ϩ 160 ϭ 1,550 ϩ 160 ϭ 1,710 votes.
Check Yourself 8
The Randolphs used 12 more gallons (gal) of fuel oil in October than
in September and twice as much oil in November as in September. If
they used 132 gal for the 3 months, how much was used each month?
CHAPTER 11
Page 760
An Introduction to Algebra
Check Yourself ANSWERS
1. (a) x ϭ Ϫ4; (b) x ϭ 3
2. x ϭ 3
3. (a) x ϭ Ϫ12; (b) x ϭ 24
5
4. x ϭ
5. x ϭ 6
6. x ϭ 5
7. x ϭ Ϫ14
2
8. 30 gal in September, 42 gal in October, 60 gal in November
b
Reading Your Text
The following fillintheblank exercises are designed to ensure that you
understand some of the key vocabulary used in this section.
SECTION 11.6
(a) The first goal for solving an equation is to
term on one side of the equation.
(b) Apply the
property.
the variable
property before applying the multiplication
(c) Always return to the
(d) An equation in the form x ϭ
Apago PDF Enhancer
equation to check your result.
or
ϭ x has been
.
Basic Mathematical Skills with Geometry
8:56 AM
The Streeter/Hutchison Series in Mathematics
760
9/7/09
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Page 761
Challenge Yourself

Calculator/Computer

Career Applications

Above and Beyond
< Objective 1 >
Boost your GRADE at
ALEKS.com!
Solve and check.
1. 2x ϩ 1 ϭ 9
11.6 exercises
2. 3x Ϫ 1 ϭ 17
> Videos
3. 3x Ϫ 2 ϭ 7
4. 5x ϩ 3 ϭ 23
• Practice Problems
• SelfTests
• NetTutor
• eProfessors
• Videos
Name
5. 4x ϩ 7 ϭ 35
6. 7x Ϫ 8 ϭ 13
Section
7. 2x ϩ 9 ϭ 5
8. 6x ϩ 25 ϭ Ϫ5
9. 4 Ϫ 7x ϭ 18
© The McGrawHill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
11. 3 Ϫ 4x ϭ Ϫ9
13.
15.
x
ϩ1ϭ5
2
10. 8 Ϫ 5x ϭ Ϫ7
> Videos
12. 5 Ϫ 4x ϭ 25
14.
> Videos
x
Ϫ5ϭ3
4
16.
x
Ϫ2ϭ3
3
x
ϩ3ϭ8
5
Apago PDF Enhancer
2
17. x ϩ 5 ϭ 17
3
3
18. x Ϫ 5 ϭ 4
4
4
19. x Ϫ 3 ϭ 13
5
5
20. x ϩ 4 ϭ 14
7
21. 5x ϭ 2x ϩ 9
> Videos
22. 7x ϭ 18 Ϫ 2x
23. 3x ϭ 10 Ϫ 2x
24. 11x ϭ 7x ϩ 20
25. 9x ϩ 2 ϭ 3x ϩ 38
26. 8x Ϫ 3 ϭ 4x ϩ 17
27. 4x Ϫ 8 ϭ x Ϫ 14
29. 5x ϩ 7 ϭ 2x Ϫ 3
Date
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
28. 6x Ϫ 5 ϭ 3x Ϫ 29
> Videos
30. 9x ϩ 7 ϭ 5x Ϫ 3
31. 7x Ϫ 3 ϭ 9x ϩ 5
32. 5x Ϫ 2 ϭ 8x Ϫ 11
33. 5x ϩ 4 ϭ 7x Ϫ 8
34. 2x ϩ 23 ϭ 6x Ϫ 5
SECTION 11.6
761
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11.6 exercises
Answers
35.
36.
37.
38.
39.
40.
< Objective 2 >
35. 2x Ϫ 3 ϩ 5x ϭ 7 ϩ 4x ϩ 2
36. 8x Ϫ 7 Ϫ 2x ϭ 2 ϩ 4x Ϫ 5
37. 6x ϩ 7 Ϫ 4x ϭ 8 ϩ 7x Ϫ 26
38. 7x Ϫ 2 Ϫ 3x ϭ 5 ϩ 8x ϩ 13
39. 9x Ϫ 2 ϩ 7x ϩ 13 ϭ 10x Ϫ 13
40. 5x ϩ 3 ϩ 6x Ϫ 11 ϭ 8x ϩ 25
41. 8x Ϫ 7 ϩ 5x Ϫ 10 ϭ 10x Ϫ 12
42. 10x Ϫ 9 ϩ 2x Ϫ 3 ϭ 8x Ϫ 18
43. SOCIAL SCIENCE There were 55 more yes votes than no votes on an election
measure. If 735 votes were cast in all, how many yes votes were there?
44. BUSINESS AND FINANCE Juan worked twice as many hours as Jerry. Marcia
41.
worked 3 more hours than Jerry. If they worked a total of 31 hours, how
many hours did each employee work?
42.
45. BUSINESS AND FINANCE Francine earns $120 per month more than Rob. If they
earn a total of $2,680 per month, how much does Francine earn each month?
43.
Basic Skills
46.

Challenge Yourself
 Calculator/Computer  Career Applications

Above and Beyond
Apago PDF Enhancer
Solve each equation.
47.
48.
49.
50.
51.
47. 7(2x Ϫ1) Ϫ 5x ϭ x ϩ 25
48. 9(3x ϩ 2) Ϫ 10x ϭ 12x Ϫ 7
49. 3x ϩ 2(4x Ϫ 3) ϭ 6x Ϫ 9
50. 7x ϩ 3(2x ϩ 5) ϭ 10x ϩ 17
51.
8
2
x Ϫ 3 ϭ x ϩ 15
3
3
52.
3
12
x ϩ 7 ϭ 31 Ϫ x
5
5
53.
2x
12x
Ϫ5ϭ
ϩ8
5
5
54.
24x
3x
Ϫ5ϭ
ϩ7
7
7
55. 5.3x Ϫ 7 ϭ 2.3x ϩ 5
52.
Basic Skills  Challenge Yourself  Calculator/Computer 
53.
56. 9.8x ϩ 2 ϭ 3.8x ϩ 20
Career Applications

Above and Beyond
57. AGRICULTURAL TECHNOLOGY The estimated yield Y of a field of corn (in
54.
bushels per acre) can be found by multiplying the rainfall r, in inches, during
the growing season by 16 and then subtracting 15. This relationship can be
modeled by the formula
55.
Y ϭ 16r Ϫ 15
56.
If a farmer wants a yield of 159 bushels per acre, then we can write the
equation shown to determine the amount of rainfall required.
159 ϭ 16r Ϫ 15
How much rainfall is necessary to achieve a yield of 159 bushels of corn
per acre?
57.
762
SECTION 11.6
The Streeter/Hutchison Series in Mathematics
45.
© The McGrawHill Companies. All Rights Reserved.
during aerobic training, subtract the person’s age from 220, and then
9
multiply the result by . Determine the age of a person if the person’s upper
10
limit heart rate is 153.
44.
Basic Mathematical Skills with Geometry
46. SCIENCE AND MEDICINE To determine the upper limit for a person’s heart rate
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11.6 exercises
58. CONSTRUCTION TECHNOLOGY The number of studs s required to build a wall
(with studs spaced 16 inches on center) is equal to the one more than
the length of the wall w, in feet. We model this with the formula
3
times
4
3
sϭ wϩ1
4
58.
If a contractor uses 22 studs to build a wall, how long is the wall?
59. ALLIED HEALTH The internal diameter D [in millimeters (mm)] of an endotra
cheal tube for a child is calculated using the formula
Dϭ
Answers
t ϩ 16
4
59.
60.
61.
in which t is the child’s age (in years).
How old is a child who requires an endotracheal tube with an internal
diameter of 7 mm?
62.
63.
© The McGrawHill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
60. MECHANICAL ENGINEERING The number of BTUs required to heat a house is
3
2 times the volume of the air in the house (in cubic feet). What is the maxi4
mum air volume that can be heated with a 90,000BTU furnace?
Basic Skills

Challenge Yourself

Calculator/Computer

Career Applications

64.
Above and Beyond
Apago PDF Enhancer
61. Create an equation of the form ax ϩ b ϭ c that has 2 as a solution.
62. Create an equation of the form ax ϩ b ϭ c that has Ϫ6 as a solution.
63. The equation 3x ϭ 3x ϩ 5 has no solution, whereas the equation 7x ϩ 8 ϭ 8
has zero as a solution. Explain the difference between an equation that has
zero as a solution and an equation that has no solution.
64. Construct an equation for which every real number is a solution.
Answers
1. x ϭ 4
13. x ϭ 8
3. x ϭ 3
5. x ϭ 7
7. x ϭ –2
9. x ϭ Ϫ2
11. x ϭ 3
15. x ϭ 32
17. x ϭ 18
19. x ϭ 20
21. x ϭ 3
23. x ϭ 2
25. x ϭ 6
27. x ϭ Ϫ2
33. x ϭ 6
35. x ϭ 4
37. x ϭ 5
43. 395 votes
45. $1,400
29. x ϭ Ϫ
10
3
39. x ϭ Ϫ4
47. x ϭ 4
49. x ϭ Ϫ
7
13
55. x ϭ 4
57. 10 in.
59. 12 yr
2
8
61. Above and Beyond
63. Above and Beyond
31. x ϭ Ϫ4
5
3
51. x ϭ 9
41. x ϭ
3
5
53. x ϭ Ϫ
SECTION 11.6
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summary :: chapter 11
Example
Section 11.1
The sum of x and 5 is x ϩ 5.
7 more than a is a ϩ 7.
b increased by 3 is b ϩ 3.
Subtraction x Ϫ y means the difference of x and y, or x
minus y. Some other words indicating subtraction are less
than and decreased by.
The difference of x and 3 is x Ϫ 3.
5 less than p is p Ϫ 5.
a decreased by 4 is a Ϫ 4.
Multiplication
The product of m and n is mn.
The product of 2 and the sum of a and
b is 2(a ϩ b).
x#y
(x)(y) s These all mean the product of x and y, or x times y.
xy
x
means x divided by y, or the quotient when x is
y
divided by y.
Division
n
n divided by 5 is . The sum of a
5
aϩb
and b, divided by 3, is
.
3
Evaluating Algebraic Expressions
Step 1
Step 2
Replace each variable with the given number value.
Do the necessary arithmetic operations, following
the rules for the order of operations.
p. 703
Section 11.2
Apago D
P Evaluate
F Enhancer
4a Ϫ b
p. 711
2c
if a ϭ Ϫ6, b ϭ 8, and c ϭ Ϫ4.
4(Ϫ6) Ϫ (8)
4a Ϫ b
ϭ
2c
2(Ϫ4)
Ϫ24 Ϫ 8
ϭ
Ϫ8
Ϫ32
ϭ4
ϭ
Ϫ8
Simplifying Algebraic Expressions
Section 11.3
Term A number, or the product of a number and one or
more variables, raised to a power.
4a2 and 3a2 are like terms.
Like terms Terms that contain exactly the same variables
raised to the same powers.
5x2 and 2xy2 are not like terms.
p. 721
Combining Like Terms
Step 1
Add or subtract the numerical coefficients.
Step 2
Attach the common variables.
764
5a ϩ 3a ϭ 8a
7xy Ϫ 3xy ϭ 4xy
p. 723
Basic Mathematical Skills with Geometry
Addition x ϩ y means the sum of x and y, or x plus y.
Some other words indicating addition are more than and
increased by.
The Streeter/Hutchison Series in Mathematics
From Arithmetic to Algebra
Reference
© The McGrawHill Companies. All Rights Reserved.
Definition/Procedure
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summary :: chapter 11
Definition/Procedure
Example
Reference
Using the Addition Property to Solve an Equation
Section 11.4
Equation A statement that two expressions are equal.
3x Ϫ 5 ϭ 7 is an equation.
Solution Any value for the variable that makes an equation
a true statement.
4 is a solution to the equation
because
The Streeter/Hutchison Series in Mathematics
Basic Mathematical Skills with Geometry
3(4) Ϫ 5 7
12 Ϫ 5 7
7ϭ7
(True)
Equivalent equations Equations that have exactly the
same set of solutions.
3x Ϫ 5 ϭ 7 and x ϭ 4 are equivalent
equations.
p. 732
The addition property If a ϭ b, then a ϩ c ϭ b ϩ c.
Adding (or subtracting) the same quantity to both sides of
an equation yields an equivalent equation.
xϪ5ϭ 7
ϩ 5 ϩ5
x
ϭ 12
p. 732
Using the Multiplication Property to Solve an Equation
Section 11.5
The multiplication property If a ϭ b and c 0, then
ac ϭ bc. Multiplying (or dividing) both sides of an equation
by the same nonzero number yields an equivalent equation.
5x ϭ 20
20
5x
ϭ
5
5
xϭ4
p. 743
To solve a percent problem algebraically, translate the
problem into algebra (writing the rate as a decimal) and use
the multiplication rule to solve.
30% of what number is 45?
p. 747
Apago PDF Enhancer
0.3x ϭ 45
0.3x
45
ϭ
0.3
0.3
x ϭ 150
Combining the Properties to Solve Equations
© The McGrawHill Companies. All Rights Reserved.
p. 730
Solving linear equations We say that an equation is solved
when we have an equivalent equation of the form x ϭ or
ϭ x in which is some number.
The steps for solving a linear equation follow.
Step 1
Use the distributive property to remove any
grouping symbols.
Step 2
Combine like terms on each side of the equation.
Step 3
Add or subtract variable terms to bring the variable
term to one side of the equation.
Step 4
Add or subtract numbers to isolate the variable term.
Step 5
Multiply by the reciprocal of the coefficient to
isolate the variable.
Step 6
Check your result.
Section 11.6
Solve:
3x Ϫ 6 ϩ 4x
7x Ϫ 6
7x Ϫ 6 Ϫ 3x
4x Ϫ 6
4x Ϫ 6 ϩ 6
4x
4x
4
x
p. 759
ϭ
ϭ
ϭ
ϭ
ϭ
ϭ
3x ϩ 14
3x ϩ 14
3x ϩ 14 Ϫ 3x
14
14 ϩ 6
20
20
ϭ
4
ϭ5
Check:
3(5) Ϫ 6 ϩ 4(5) 3(5) ϩ 14
29 ϭ 29 True
765