Section 8.1
Solving Linear Equations:
x + b = c and ax = c
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Objectives
o Define the term “linear equation.”
o Solve equations of the form x + b = c.
o Solve equations of the form ax = c.
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Linear Equations
Linear Equation in x
If a, b, and c are constants and a  0 , then a linear
equation in x is an equation that can be written in the
form
ax + b = c.
Note: A linear equation in x is also called a first-degree
equation in x because the variable x can be written
with the exponent 1. That is, x = x1.
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Solving Equations of the Form x + b = c
Addition Principle of Equality
If the same algebraic expression is added to both sides
of an equation, the new equation has the same
solutions as the original equation. Symbolically, if A, B,
and C are algebraic expressions, then the equations
A=B
and
A+C=B+C
have the same solutions.
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Solving Equations of the Form x + b = c
Procedure for Solving Linear Equations that Simplify
to the Form x + b = c
1. Combine any like terms on each side of the
equation.
2. Use the addition principle of equality and add the
opposite of the constant b to both sides. The
objective is to isolate the variable on one side of
the equation (either the left side or the right side)
with a coefficient of +1.
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Solving Equations of the Form x + b = c
Procedure for Solving Linear Equations that Simplify
to the Form x + b = c (cont.)
3. Check your answer by substituting it for the
variable in the original equation.
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Example 1: Solving x + b = c
Solve the equation x - 3 = 7.
Write the equation.
Solution
x -3= 7
x -3+ 3= 7+ 3
x = 10
Check
x -3= 7
?
10 - 3 = 7
7=7
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Add 3 (the opposite of -3) to both sides.
Simplify.
Substitute x = 10.
true statement
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Example 2: Solving x + b = c
Solve the equation -11 = y + 5.
Write the equation.
Solution -11 = y + 5
Note that the
variable can be on the right side.
-11 - 5 = y + 5 - 5
-16 = y
Check
Add -5 (the opposite of + 5) to both sides.
Simplify.
-11 = y + 5
?
-11 =  -16  + 5 Substitute y = -16.
-11 = -11
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true statement
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Example 3: Solving x + b = c
Solve the equation -4.1 + x = 0.
Solution -4.1 + x = 0
Write the equation.
-4.1 + x + 4.1 = 0 + 4.1
x = 4.1
-4.1 + x = 0
?
-4.1 +  4.1 = 0


Add 4.1 the opposite of -4.1 to both sides.
Simplify.
Check
0=0
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Substitute x = 4.1.
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Example 4: Solving x + b = c
2 3
Solve the equation x - = .
5 10
2 3
Solution x - =
Write the equation.
5 10
2 2 3 2
2
2
x- + = +
Add
the opposite of - to both sides.
5
5
5 5 10 5
3 4
x= +
Simplify.  The common denominator is 10.
10 10
7
x=
Simplify.
10

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
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Example 4: Solving x + b = c (cont.)
2 3
x- =
Check
5 10
 7  2? 3
  - =
10
5 10
7 4 ? 3
- =
10 10 10
3
3
=
10 10
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Substitute x =
7
.
10
The common denominator is 10.
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Example 5: Simplifying and Solving Equations
Simplify and solve the equation 3z - 2z + 2 = 3 + 8.
Solution 3z - 2z + 2 = 3 + 8
z + 2 = 11
z + 2 - 2 = 11 - 2
z=9
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Write the equation.
Combine like terms on
both sides of the equation.
Add -2 (the opposite of +2)
to both sides.
Simplify.
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Example 5: Simplifying and Solving Equations
(cont.)
3z - 2z + 2 = 3 + 8
?
3 9 - 2  9 + 2 = 3 + 8
?
27 - 18 + 2 = 3 + 8
Check
11 = 11
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Substitute z = 9.
Simplify.
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Completion Example 6: Simplifying and
Solving Equations
Supply the reasons for each step in solving the
equation.
5x - 4 x - 1.5 = 6.3 + 4.0
Solution
Write the equation.
5x - 4 x - 1.5 = 6.3 + 4.0 __________________________
x - 1.5 = 10.3
x - 1.5 + 1.5 = 10.3 + 1.5
x = 11.8
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Combine like terms on both sides
of the equation.
__________________________
Add 1.5 (the opposite of - 1.5) to
__________________________
both sides of the equation.
Simplify.
________________
__________
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Solving Equations of the Form ax = c
Multiplication (or Division) Principle of Equality
If both sides of an equation are multiplied by (or
divided by) the same nonzero constant, the new
equation has the same solutions as the original
equation. Symbolically, if A and B are algebraic
expressions and C is any nonzero constant, then the
equations
A=B
and AC = BC where C  0
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Solving Equations of the Form ax = c
Multiplication (or Division) Principle of Equality (cont.)
A B
and = where C  0
C C
have the same solutions. We say that the equations
are equivalent.
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Solving Equations of the Form ax = c
Procedure for Solving Linear Equations that Simplify
to the Form ax = c
1. Combine any like terms on each side of the
equation.
2. Use the multiplication (or division) principle of
equality and multiply both sides of the equation by
the reciprocal of the coefficient of the variable.
(Note: This is the same as dividing both sides of the
equation by the coefficient.) Thus the coefficient of
the variable will become +1.
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Solving Equations of the Form ax = c (cont.)
Procedure for Solving Linear Equations that Simplify
to the Form ax = c (cont.)
3. Check your answer by substituting it for the
variable in the original equation.
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Example 7: Solving ax = c
Solve the equation 5x = 20.
Solution
5x = 20
Write the equation.
1
1
  5x  =  20
5
5
1 20
1 
  5 x = 
5
5 1
1 x = 4
x=4
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1
Multiply by , the reciprocal of 5.
5
Use the associative property of multiplication.
Simplify.
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Example 7: Solving ax = c (cont.)
Check
5x = 20
?
5   4  = 20
20 = 20
Substitute x = 4.
true statement
Multiplying by the reciprocal of the coefficient is the
same as dividing by the coefficient itself. So, we can
1
multiply both sides by as we did, or we can divide
5
both sides by 5.
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Example 7: Solving ax = c (cont.)
In either case, the coefficient of x becomes +1.
5x = 20
5x 20
=
5
5
x=4
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Divide both sides by 5.
Simplify.
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Example 8: Solving ax = c
Solve the following equation.
1.1x + 0.2x = 12.2 − 3.1
Solution
When decimal coefficients or constants are involved,
you might want to use a calculator to perform some of
the arithmetic.
1.1x + 0.2x = 12.2 - 3.1 Write the equation.
Combine like terms.
1.3x = 9.1
1.3x 9.1
Divide both sides by 1.3.
=
1.3 1.3
x = 7.0
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Example 8: Solving ax = c (cont.)
Check
1.1x + 0.2x = 12.2 - 3.1
?
1.1 7 + 0.2  7 = 12.2 - 3.1
?
7.7 + 1.4 = 9.1
9.1 = 9.1
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Substitute x = 7.
Simplify.
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Example 9: Solving ax = c
Solve the following equation.
−x = 4
Solution
-x = 4
Write the equation.
-1x = 4
- 1 is the coefficient of x .
-1x 4
=
-1 -1
Divide by - 1 so that the
coefficient will become + 1.
x = -4
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Example 9: Solving ax = c (cont.)
Check
-x = 4
?
-  -4  = 4
Substitute x = -4.
4=4
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Completion Example 10: Solving ax = c
Supply the reasons for each step in solving the
following equation.
4x 3
=
5 10
4
3
4
4x
(This could be written x =
because x is the same as
.)
5
10
5
5
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Completion Example 10: Solving ax = c (cont.)
4x 3
=
Solution
5 10
5 4
5 3
 x= 
4 5
4 10
1 5 3
1 x =

4 2 5
3
x=
8
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Write the equation.
___________________
5
Multiply both sides by .
4
___________________
Simplify.
___________________
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Example 11: Application
Mark had a coupon for $50 off the Blu-Ray player he
just purchased. With the coupon, the price was
$165.50. Solve the equation y − 50 = 165.50 to
determine the original price of the Blu-Ray player.
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Example 11: Application (cont.)
Solution
y − 50 = 165.50
y −50 + 50 = 165.50 + 50
y = 215.50
Use the addition principle by
adding 50 to both sides.
Simplify.
The original price of the Blu-Ray player was $215.50.
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Practice Problems
Solve the following equations.
1. -16 = x + 5 2. 6y - 1.5 = 7.5
3. 4x = -20
3y
= 33
4.
5
5. 1.7z + 2.4z = 8.2
7. 3x = 10
4
8. 5x - 4x + 1.6 = -2.7 9. = 3x
5
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6. -x = -8
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Practice Problem Answers
1. x = -21
2. y = 1.5
3. x = - 5
4. y = 55
5. z = 2
6. x = 8
8. x = - 4.3
4
=x
9.
15
10
7. x =
3
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