
ExamplesExample 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. 