Solving Multi-step Equations

Learning Objective(s)

·         Simplify algebraic equations using the Properties of Equality and the Distributive Property to clear parentheses and fractions.

Introduction

There are some equations that you can solve in your head, in an instant. If I asked you to solve 2y = 4, chances are you wouldn’t need to get out a pencil and paper to calculate that y = 2. You only needed to do one thing to get the answer, divide by 2.

Other equations are more complicated. Can you solve  without writing anything down? Me neither. That’s because it contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several operations to solve. Don’t worry though. Although they take more time and more operations, multi-step equations can still be simplified and solved by applying basic algebraic rules. Two helpful tools are the Distributive Property and multiplication by a common denominator.

Removing Parentheses with the Distributive Property

Just as with simpler equations, solving multi-step equations usually means isolating a variable on one side of the equals sign.  Step-by-step, we must get the variable out of parentheses, away from other terms, and with a coefficient of 1. If a variable is inside parentheses, they can be cleared by applying the Distributive Property of Multiplication.

The Distributive Property states that for all real numbers a, b, and c, . What that means is that when a number multiplies an expression inside parentheses, we can distribute the multiplication to each term of the expression individually. Let’s go back to the equation . We can apply the distributive property and clear the parentheses by multiplying each term inside of them by 6. The expression  then becomes . Now there are no parentheses.

In which of the following equations is the distributive property properly applied to the equation ?

A)

B)

C)

D)

Clearing Fractions with Common Denominators

We’ve seen how to simplify a multi-step equation by applying the Distributive Property to clear parentheses. It’s also helpful to simplify equations that include fractions before we try to solve them.  We can do this by multiplying both sides by a common denominator of all of the fractions in the equation.

Let’s go back to the equation . The lowest common denominator of the fractions  and  is 8.  So we’ll multiply the expression with those fractions by 8.  Because we must preserve the equality of the equation, we’ll multiply the expression on the other side by 8, too. Now we’ve got .

But wait—we just put parentheses back into the equation again. Now what? Simple, we go back to our old friend the Distributive Property once again. Once we multiply each term inside the parentheses by 8, both the parentheses and the fractions will disappear.  Sweet.

In order to clear away the fractions from , we can multiply both sides of the equation by which of the following numbers?

3   6   9  18

A) 6

B) 3 and 6

C) 9

D) 6 and 18