Section 2.4
Solving Linear Equations:
ax + b = c
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Objective
o Solve equations of the form ax + b = c.
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Solving Equations of the Form ax + b = c
Procedure for Solving Linear Equations that Simplify
to the Form ax + b = c
1. Combine like terms on both sides of the equation.
2. Use the addition principle of equality and add the
opposite of the constant b to both sides.
3. Use the multiplication (or division) principle of
equality to multiply both sides by the reciprocal of
the coefficient of the variable (or divide both sides
by the coefficient itself). The coefficient of the
variable will become +1.
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Solving Equations of the Form ax + b = c
Procedure for Solving Linear Equations that Simplify
to the Form ax + b = c (cont.)
4. Check your answer by substituting it for the variable
in the original equation.
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Example 1: Solving Linear Equations
Solve each of the following equations.
a. 3x + 3 = -18
Solution 3x + 3 = -18
Write the equation.
3x + 3 - 3 = -18 - 3
3x = -21
3x -21
=
3
3
x = -7
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Add -3 to both sides.
Simplify.
Divide both sides by 3.
Simplify.
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Example 1: Solving Linear Equations (cont.)
Check:
3x + 3 = -18
?
3  -7 + 3 = -18
?
-21 + 3 = -18
-18 = -18
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Substitute x = -7.
Simplify.
True statement
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Example 1: Solving Linear Equations (cont.)
b. –26 = 2y – 14 – 4y
Solution
-26 = 2y - 14 - 4 y
Write the equation.
-26 = -2y - 14
Combine like terms.
-26 + 14 = -2y - 14 + 14
Add 14 to both sides.
-12 = -2y
Simplify.
-12 -2y
=
-2
-2
Divide both sides by -2.
6=y
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Simplify.
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Example 1: Solving Linear Equations (cont.)
Check: -26 = 2y - 14 - 4 y
?
-26 = 2  6  - 14 - 4  6 
Substitute y = 6.
?
-26 = 12 - 14 - 24
Simplify.
-26 = -26
True statement
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Example 2: Solving Linear Equations
Involving Decimals
Solve each of the following equations.
a. 16.53 – 18.2z – 7.43 = 0
Solution
Write the equation.
16.53 - 18.2z - 7.43 = 0
100 16.53 - 18.2z - 7.43 = 100  0
1653 - 1820z - 743 = 0
910 - 1820z = 0
910 - 1820z - 910 = 0 - 910
-1820z = -910
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Multiply both sides by 100.
(This results in integer
coefficients.)
Simplify.
Combine like terms.
Add -910 to both sides.
Simplify.
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Example 2: Solving Linear Equations
Involving Decimals (cont.)
-1820 z -910
=
-1820 -1820
1

z = 0.5  or z = 

2
Divide both sides by -1820.
Simplify.
Check: 16.53 - 18.2z - 7.43 = 0
?
16.53 - 18.2  0.5 - 7.43 = 0
?
16.53 - 9.10 - 7.43 = 0
0=0
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Substitute z = 0.5.
Simplify.
True statement
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Example 2: Solving Linear Equations
Involving Decimals (cont.)
b. 5.1x + 7.4 – 1.8x = -9.1
Solution
5.1x + 7.4 - 1.8 x = -9.1
10  5.1x + 7.4 - 1.8 x  = 10  -9.1
51x + 74 - 18 x = -91
33x + 74 = -91
33x + 74 - 74 = -91 - 74
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Write the equation.
Multiply both sides by 10.
(This results in integer
coefficients.)
Simplify.
Combine like terms.
Add -74 to both sides.
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Example 2: Solving Linear Equations
Involving Decimals (cont.)
33x = -165
Simplify.
33x -165
=
33
33
Divide both sides by the coefficient 33.
x = -5
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Simplify.
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Example 3: Solving Linear Equations with
Fractional Coefficients
Solve each of the following equations.
5
5
10
a.
x- =6
2
9
5
5
10
Solution
Write the equation.
x- =6
2
9
5
5
 10 
Multiply both sides by 18 (the
18  x -  = 18  - 
LCM of the denominators).
6
 9
2
5 
 5
 10 
18  x  - 18   = 18  - 
6 
 2
 9
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Apply the distributive
property.
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Example 3: Solving Linear Equations with
Fractional Coefficients (cont.)
15x - 45 = -20
15x - 45 + 45 = -20 + 45
Simplify.
Add 45 to both sides.
15x = 25
Simplify.
15x 25
=
15 15
Divide both sides by 15.
5
x=
3
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Simplify.
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Example 3: Solving Linear Equations with
Fractional Coefficients (cont.)
Check:
5
5
10
x- =6
2
9
5  5  5 ? 10
  - = 6 3
2
9
25 45 ? 20
- =18 18
18
20
20
- =18
18
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5
Substitute y = .
3
Simplify.
True statement
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Example 3: Solving Linear Equations with
Fractional Coefficients (cont.)
1
3
7 2
b.
x+ x+ - x=0
2
4
2 3
1
3
7 2
Solution x + x + - x = 0
2
4
2 3
3
7 2 
1
12  x + x + - x  = 12  0
2
4
2 3 
Write the equation.
Multiply both sides by 12 (the
LCM of the denominators).
1 
3 
 7
2 
12  x  + 12  x  + 12   - 12  x  = 12  0
2 
4 
 2
3 
Apply the distributive property.
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Example 3: Solving Linear Equations with
Fractional Coefficients (cont.)
6x + 9x + 42 - 8 x = 0
7x + 42 = 0
7x + 42 - 42 = 0 - 42
Simplify.
Combine like terms.
Add -42 to both sides.
7 x = -42
Simplify.
7 x -42
=
7
7
Divide both sides by 7.
x = -6
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Simplify. Checking will show
that -6 is the solution.
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Solving Equations of the Form ax + b = c
Notes
ABOUT CHECKING
Checking can be quite time-consuming and need not be
done for every problem. This is particularly important
on exams. You should check only if you have time after
the entire exam is completed.
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Practice Problems
Solve the following linear equations.
1. x + 14 – 8x = −7
2. 2.4 = 2.6y – 5.9y – 0.9
3. n -
2n 1 1
- =
3 2 6
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4. 3x + 1 - x = 0
14 2 7
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Practice Problem Answers
1. x = 3
2. y = -1
3. n = 2
4. x = -7
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