Solving Equations

Examples

Example 6

Solve:

Solution

We can multiply both sides by any nonzero number we like. Since is the reciprocal of , we multiply each side by .

Using the multiplication principle: Multiplying both sides by “eliminates” the on the left.

1x = 8 Simplifying

x = 8 Using the identity property of 1

Check:

The solution is 8.

In Example 6, to get x alone, we multiplied by the reciprocal, or multiplicative inverse of . We then simplified the left-hand side to x times the multiplicative identity, 1, or simply x. These steps effectively replaced the on the left with 1.

Because division is the same as multiplying by a reciprocal, the multiplication principle also tells us that we can “divide both sides by the same nonzero number”. That is,

if a = b then (provided c 0 ).

In a product like 3x , the multiplier 3 is called the coefficient. When the coefficient of the variable is an integer or a decimal, it is usually easiest to solve an equation by dividing on both sides. When the coefficient is in fraction notation, it is usually easier to multiply by the reciprocal.

Example 7

Solve:

Solution

a)

Using the multiplication principle: Dividing both sides by -4 is the same as multiplying by

1x = -23 Simplifying

x = -23 Using the identity property of 1

Check:

The solution is -23.

b)

Dividing both sides by 3 or multiplying both sides by

Simplifying

Check:

The solution is 4.2.

c) To solve an equation like -x = 9 remember that when an expression is multiplied or divided by -1 its sign is changed. Here we multiply both sides by -1 to change the sign of -x :

-x = 9

(-1)(-x) = (-1)9 Multiplying both sides by -1 (Dividing by -1 would also work)

x = -9 Note that (-1)(-x) is the same as (-1)(-1)x

Check:

The solution is -9.

d) To solve an equation like we rewrite the left-hand side as and then use the multiplication principle:

Rewriting as

Multiplying both sides by

Removing a factor equal to

y = 12

Check:

The solution is 12.