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
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.
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.
Dividing 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.