# 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. |