Solving Multistep Algebraic Equations Study Guide

Introduction to Solving Multistep Algebraic Equations

Each problem that I solved became a rule which served afterwards to solve other problems.

—Rene Descartes (1596–1650) French Philosopher and Mathematician

In this lesson, you'll learn how to use more than one operation to solve algebraic equations.

We often need to use more than one step to solve an equation. We might have to add and then divide, or add, subtract, and then multiply to find the value of a variable.

Example

4x + 2 = 22

Subtraction alone will not give us the value of x. Division alone will not give us the value of x, either. We must perform two operations. When you are solving an equation that requires more than one step, the biggest question is: Which operation should be done first? The good news is, it does not matter. Whether we divide first and then subtract, or subtract and then divide, we will still arrive at the same answer. However, it is usually easier to perform operations in the reverse sequence from the "order of operations." After simplifying, start with any addition or subtraction, then move on to any multiplication or division (and then deal with any exponents, if needed).

Let's start by using subtraction. Subtract 2 from both sides of the equation:

4x + 2 – 2 = 22 – 2

4x = 20

Now, the equation looks like the equations we solved in the previous lesson. Divide both sides of the equation by 4:

x = 5

Solve that equation again, only this time, start with division:

4x + 2 divided by 4 is , or , and 22 divided by 4 is , or To finish solving the equation, subtract from both sides of the equation:

x = 5

We found the same answer, x = 5, but you might have felt that the problem was a little tougher to solve if you do not like working with fractions. Often, if you divide before adding or subtracting, you will have to work with fractions in order to solve an equation.

Example

In this problem, the variable g is divided by 6, and 5 is subtracted from that quotient. We can either multiply and then add, or add and then multiply. Let's try multiplying first. We must multiply both terms on the left side of the equation:

g – 30 = 42

Now, we can add 30 to both sides of the equation to solve for g

g – 30 + 30 = 42 + 30

g = 72

Would adding before multiplying have made the problem easier to solve? That's up to you. We could have added 5 to both sides of the equation, which would have given us These numbers are a little smaller than 30 and 42, which were the numbers we had after multiplying. Often, if you add or subtract before multiplying, you will have smaller numbers with which to work.

One exception to adding or subtracting first is when an expression on either side of the equal sign can be simplified, such as when a constant or variable is multiplying an expression in parentheses. In that case, it is better to use the distributive law first. Why? You may find that after multiplying, you can combine like terms and make the equation a little easier to solve.