Download Section 2.2 The Multiplication Property of Equality

Math 123 – Section 2.2 - The Multiplication Property of Equality - Part I - Page 1
Section 2.2
The Multiplication Property of Equality - Part I
I. Definitions
A. A linear or first-degree equation is one in which the variable occurs in only the first
power.
B. Examples
1. 5x – 7 = 38
2. 5(x + 6) – 3(x – 2) = 14(3x – 7) + 8x
C. A linear equation in standard form will look like
Ax + B = 0,
Where A is not 0.
II. Properties to use Solving Linear Equations
A. Addition Property
1. Add (subtract) the same number from both sides of the equation.
2. Examples
a. 7x – 5 = 19
Add 5 to both sides to get:
7x – 5 + 5 = 19 + 5
Or
7x = 24
b. 26x + 15 = 45x – 18
Subtract 26x to both sides to get:
26x + 15 – 26x = 45x – 18 – 26x
Or
15 = 19x – 18
B. Multiplication Property
1. Multiply (divide) the same nonzero number from both sides of the equation.
2. Examples
x
a.
=7
5
Multiply both sides by 5 to get:
x
5   = 5(7)
5
Or
x = 35
b. 25x = -175
Divide both sides by 25 to get:
25x
175
=25
25
or
x = -7
© Copyright 2009 by John Fetcho. All rights reserved
Math 123 – Section 2.2 - The Multiplication Property of Equality - Part I - Page 2
III. Solving Linear Equations
A. Goal
1. We want to get each equation to look like:
Variable = Constant
2. Examples
a. y = -17
11
b.
=x
3
B. Procedure
1. Simplify each side of the equation by combining like terms (but just on one
side or the other, don’t combine across the equal sign!).
2. Get all the variable terms together on one side of the equation by using the
Addition Property.
3. Get all the constant terms together on one side of the equation by using the
Addition Property.
4. Isolate the variable by using the Multiplication Property.
5. Check your answer in the original equation.
C. Examples – Solve each of the following.
1. 5w + 7w – 2 = 5 + 7 – 2w
First, combine like terms on both sides of the equation to get:
12w – 2 = 12 – 2w
Next, add 2w to both sides to get:
12w – 2 + 2w = 12 – 2w + 2w
or
14w – 2 = 12
Then add 2 to both sides to get:
14w – 2 + 2 = 12 + 2
or
14w = 14
Finally, divide both sides by 14 to get:
14w 14
=
14
14
Answer: w = 1
Check:
5(1) + 7(1) – 2 = 5 + 7 – 2(1)
5 + 7 – 2 = 12 – 2
10 = 10
2. – x = 1
Remember that “– x” really means –1x, so to isolate x we need to divide both
sides by –1
−x
1
=
−1 −1
OR
Answer: x = – 1
© Copyright 2009 by John Fetcho. All rights reserved
Math 123 – Section 2.2 - The Multiplication Property of Equality - Part I - Page 3
3. 3x + 7 = 5x + 6
Subtract 3x from both sides to get:
3x + 7 – 3x = 5x + 6 – 3x
or
7 = 2x + 6
Next, subtract 6 from both sides to get:
7 – 6 = 2x + 6 – 6
or
1 = 2x
Now divide both sides by 2 to get:
1 2x
=
2
2
OR
1
Answer:
=x
2
4. Now you try one: –2x – 7x + x = –12 – 12
Answer: x = 3
5.
x
=4
7
This is really a division problem. What it means is "x divided by 7 is 4." What
is the opposite operation? So let's multiply both sides by 7 to get:
x
7   = 7 ( 4)
7
Answer:
6.
Cancel out the 7's on the left-hand side.
28 = x
Now you try one:
Answer:
−
x
= −9
5
x = 45
© Copyright 2009 by John Fetcho. All rights reserved