We can multiply both sides by any nonzero number we like.
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
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
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.
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
The solution is -23.
both sides by 3 or multiplying both sides by
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
The solution is -9.
d) To solve an equation like we
rewrite the left-hand side as and then
use the multiplication principle:
Multiplying both sides by
Removing a factor equal to
y = 12
The solution is 12.