2.3a
The Multiplication Principle of Equality
1. Solve linear equations using the
multiplication principle.
2. Solve linear equations using both the
addition and the multiplication principles.
Review
1
1
4 5 1
  1
51 41 1
Multiplicative Inverses
(Reciprocals)
6
2  18  2  12

 12
18     
3
1  3 1 1
2

18   12
 3 1
6
Multiplication Principle of Equality
44
24   24 
88
10  10
1
1
10   10 
2
2
55
33
 23   23
 6  6
10  10
10 10

2
2
55
Multiplying by ½ and
dividing by 2 are
equivalent!
The Multiplication Principle of Equality
If a = b, then ac = bc for all real numbers a, b, and
c, where c  0.
Multiplication Principle of Equality

Multiply both sides of the equation by the
multiplicative inverse of the coefficient
or

Divide both sides by the coefficient.
8x  16
1 1
1 2
8 x   16
81
1 8
1x  2
x2
1
16 2
8x  16

81 8 1
1x  2
x2
If the coefficient is a fraction,
multiply by the multiplicative
inverse (reciprocal).
If the coefficient is an integer,
divide.
Multiplication Principle of Equality
1
7 x  15

71 7
15
x
7
1
1
-3
65 
6
 15
 y 15
51 6 1 
51
 18
y
1
y  18
Addition and Multiplication Principle of Equality
4 x  7  23
To Solve Linear Equations
1. Simplify both sides of the equation as needed.
a. Distribute to clear parentheses.
b. Combine like terms.
2. Use the addition principle so that all variable terms
are on one side of the equation.
3. Use the addition principle so that all constants are
on the other side.
4. Use the multiplication principle to isolate the
variable.
Use the addition principle BEFORE the multiplication principle!!!
Addition and Multiplication Principle of Equality
4 x  7  23
7 7
4 x  16
4
4
x4
Check:
44   7  23
16  7  23
23  23
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Addition and Multiplication Principle of Equality
 5x  12  2x  7
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
 3x  12  7
 12  12
 3x  5
3 3
5
x
3
Check:
5
5
 5   12  2   7
3
3
 25
10
 12 
7
3
3
 15
 12  7
3
 5  12  7
77
Addition and Multiplication Principle of Equality
 3m  4   25  2m   29
 3m  12  10  4m  29
m  22  29
 22  22
m7
Check:
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Addition and Multiplication Principle of Equality
2 x  5  3  3x  9
2x  10  3  3x  9
2x  7  3x  9
 3x
 3x
5x  7  9
7 7
5x  16
5 5
16
x
5
Check:
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Addition and Multiplication Principle of Equality
2  17  5m   9m  m  7
2  17  5m  9m  m  7
 15  5m  8m  7
 5m  5m
 15  3m  7
7
7
 8  3m
3 3
8
 m
3
8
m
3
Check
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Addition and Multiplication Principle of Equality
 3x  4  5x  4 x  2x  5
2x  4  2x  5
 2x
 2x
45
Contradiction
Solution:
No solution or
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Addition and Multiplication Principle of Equality
2 x  4   4 x  3  2 x  5
2x  8  2x  8
 2x
 2x
88
Identity
Solution: All Real Numbers
1. Simplify
2. Move the variable terms.
3. Move the constants.
4. Isolate the variable.
Solve. 8m + 6 = 3(12 + 2m)
a) 3
b) 5
c) 8
d) 15
2.3
Solve. 8m + 6 = 3(12 + 2m)
a) 3
b) 5
c) 8
d) 15
2.3