Basic Mathematics for College Students

Use the division property of equality.

To introduce a fourth property of equality, consider the first scale shown on the left,

which represents the equation 2x6. The scale is in balance because the weights on

the left and right sides are equal. To find x, we need to split the amount of weight on

the left side in half. To keep the scale in balance, we must split the amount of weight

in half on the right side. After doing this, we see that xis balanced by 3. Therefore,x

must be 3. We say that we have solved the equation 2x6 and that the solution is 3.

This example illustrates the following property of equality.

Division Property of Equality

Dividing both sides of an equation by the same nonzero number does not

change its solution.

For any numbers , , and , where is not 0,

if , then

When we use this property of equality, the resulting equation is equivalent to the

original one.

a

c



b

c

ab

ab c c

5

664 Chapter 8 An Introduction to Algebra

b.To isolate , we multiply both sides by the reciprocal of , which is.

This is the equation to solve.

To undo the multiplication by , multiply both sides

by the reciprocal of.

On the left side, the product of a number and its

reciprocal is 1:.

The coefficient 1 need not be written since.

The solution is ^125. Verify that this is correct by checking.

x 1 xx

12

5

1 x  (^451)  (^542)  1

12

5

^54

^4 ^54

5

a

5

4

xb

4

5

( 3 )



5

4

x 3

x ^54 ^45

EXAMPLE (^7) Solve: a. b.
StrategyWe will use a property of equality to isolate the variable on one side of
the equation.
WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form or , whose solution is obvious.
Solution
a.To isolate on the left side, we use the division property of equality. We can
undo the multiplication by 2 by dividing both sides of the equation by 2.
This is the equation to solve.
Divide both sides by 2.
On the left side, simplify by removing the common factor of 2 in the
numerator and denominator:. On the right side, do the division.
The coefficient 1 need not be written since.
If we substitute 40 for in , we obtain the true statement. This
verifies that 40 is the solution.
t 2 t 80 80  80
t 40 1 tt
2
1
t
21 t
22 t
1 t 40
2 t
2



80

2

2 t 80

t

ta number a numbert

2 t 80 6.028.6t

Self Check 7

Solve:

a.

b.

Now TryProblems 69 and 79

10.040.4r

16 x 176

2 x = 6

Split in

half

x = 3

(^1111)
x^111
x x
Split in
half
(^11)