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 lefthand 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 lefthand 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.
