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Solving Multi-Step Equations: Explanations, Review, and Examples

• The Albert Team
• Last Updated On: February 16, 2023

Whether you’re new to solving multi-step equations or simply studying before that big chapter test, Albert has you covered!

This blog post will guide you through defining multi-step equations, examples of multi-step equations, and how to solve multi-step equations (including problems with fractions and words). Let’s go!

Return to the Table of Contents

What We Review

What is a multi-step equation?

Remember, an equation is a mathematical sentence that uses an equal sign, = , to show that two expressions are equal. 

We began our study of solving equations with one-step equations , then we moved on to two-step equations . (Check out those links if you need a quick refresher!) 

Now we are moving to multi-step equations . A multi-step equation is an equation that takes two or more steps to solve. These problems can have a mix of addition, subtraction, multiplication, or division. We also might have to combine like terms or use the distributive property to properly solve our equations. 

So get your mathematical toolbox out! You never know what you might see in a multi-step equation!

Examples of multi-step equations

Multi-step equations are a wide-ranging category of equations. Some can be very simple, while others become more complex. Never fear! We’re going to show you many examples of multi-step equations and how to solve these important aspects of Algebra 1. 

Here are some examples of multi-step equations: 

How to solve multi-step equations

Remember, an equation is solved when we have isolated the variable and found a value that makes the equation true. In order to solve equations, we use inverse operations to help us isolate the variable.

Order of Operations

Another mathematical concept that will help when solving multi-step equations is the Order of Operations . To use the order of operations, we must first do any operations inside grouping symbols (parentheses, brackets, etc), then exponents, then multiplication or division (whatever comes first, left to right), then finally addition or subtraction (whatever comes first, left to right). You can remember this by the acronym, PEMDAS .

Additionally, we may have to combine like terms on either side of the equation to help solve these equations. Eventually, you will create a one- or two-step equation that you will be able to solve similarly to previous problems! 

Here is an example of a multi-step equation with variables on both sides:

Solve for x in the following equation:

Since there are variables on both sides, we must eliminate the variable from one side first. I suggest moving the 4x first, as to not create a negative. 

Now we are back to a basic two-step equation.

To check you answer, you can simplify substitute 3 into the variable to see if the equation is true: 

Thus, x = 3 is the correct solution. 

Below is a short video from Mike DeVor showing more examples of solving multi-step equations:

Now that we have been introduced to Multi-Step Equations, let’s get those brain gears in motion and look at some more challenging examples!

Multi-step equations with fractions

When dealing with an equation with more than one fraction, the easiest way to solve the equation is by finding the Least Common Denominator . The least common denominator is the smallest number that can be a common denominator for a set of fractions. 

Once we find the least common denominator, we will multiply each term by this value to eliminate the fraction. Here is an example of a multi-step equation with fractions: 

Solve for y in the following equation:

The denominators above are 2, 4, 6 , therefore the least common denominator for these numbers is 12 . So we will multiply each term by 12 .

To check your answer, you can substitute 9 into the variable to see if the equation is true:

Therefore, y = 9 is the correct solution. 

Multi-step equations with distributive property

Solve for z in the following equation:

To check you answer, you can substitute 3 into the variable to see if the equation is true:

Thus, z = 3 is the correct solution.

Solve for m in the following equation:

To check you answer, you can simplify substitute -9 into the variable to see if the equation is true:

Thus, m = -9 is the correct solution.

Multi-step equation word problems

First, let’s create an equation for the situation: 

To check you answer, you can simplify substitute 15 into the variable to see if the equation is true:

Therefore, the breakeven point for Distributor A and Distributor B would be 15  pounds.

First, let’s set up an equation that models the situation:

Since each book costs the same amount, we denote this amount by the variable, c . Then we applied the \$5 coupon to each book, and finally, we will multiply the cost of each book after the coupon by 3 . 

Now, simply solve for c like any other multi-step equation: 

Therefore, each book cost \$20 before the coupon was applied.

Keys to Remember: Solving Multi-Step Equations

• A multi-step equation is an equation that requires two or more steps to solve.
• When solving: remember whatever you do to one side, you must do to the other.
• To solve multi-step equations with fractions, you can multiply each term by the least common denominator to eliminate the fractions first.
• To check the solution, simply substitute the value into the variable to see if the equation is true.
• You can model real-life situations with an equation and solve for a correct solution.

Read these other helpful posts:

• Solving One-Step Equations
• Solving Two-Step Equations
• Forms of Linear Equations
• View ALL Algebra 1 Review Guides

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Helping with Math

Multi-Step Math Word Problems

What to expect in this article.

After reading this article, you will be able to analyze, process, and solve multi-step word problems . This lesson will provide help and guidance in solving these types of problems as it includes tips on how to solve a multi-step problem . There are two given examples wherein you can practice and guide your children in honing their mathematical skills. You can also read the common errors and misconceptions of students in solving multi-step problems. Furthermore, this article consists of links directed to worksheets – which you can find at the bottom of the page. 

What is a multi-step word problem?

Math word problems are a critical component of the mathematics curriculum because they help students develop their mental abilities , improve logical analysis , and stimulate creative thinking . Word problems are fun and challenging to solve because they represent actual situations that happen in our world. More so, having the ability to solve math word problems significantly benefits one’s career and personal life.

To be able to solve any math word problem , children must be familiar with the mathematics language associated with the mathematical symbols they are accustomed in order to comprehend the word problem.

A multi-step math word problem is a type of problem wherein you need to solve one or more problems first in order to get the necessary information to solve the question being asked. It usually involves multiple operations and may also involve more than one strand of the curriculum. Say, for example, a multi-step word problem involving area and perimeter may also require the application of ratio and multiplication .

How to solve multi-step word problems?

In any word problem, the true challenge is deciding which mathematical operation to use. In solving multi-step word problems, there may be two or more operations that you need to work on, and you must solve them in the correct order to be able to get the correct answer. Since word problems describe a real situation in detail, the question being asked can get lost in all the information, especially in a multi-step problem.

To solve multi-step word problems, you may follow these strategy:

• Analyze and understand the problem. 
• Break down each sentence of the problem and identify the clues.
• List all the information.
• Identify the unknown in the problem.
• Devise a plan or identify the mathematical operations you are going to use.
• Carry out the plan.
• Label your final answer.

Multi-Step Word Problem #1

Step 1: Break down each sentence of the problem and identify the information needed to solve the problem.

• The first sentence states that “Steven is reading a book that has 260 pages.” Hence, the total number of pages of that particular book is 260 .
• The second statement says, “He read 35 pages on Monday night and 40 pages on Tuesday night.” 

Step 2: Analyze the question of the problem and find the keyword for the unknown. The last sentence of the problem, “How many pages does he has left to read?” asks us how many more pages Steven needs to read. Hence, we are going to find the number of pages he still needs to read.

Step 3: Based on the second statement, Steven read 35 pages on a Monday night and 45 pages on a Tuesday night. Hence, we will use addition in getting the total number of pages he read for 2 nights. Thus, 

35 + 40 = 75

Therefore, Steven read 75 pages in the span of two days. However, that is not the answer we are looking for. 

Step 4: Since we are asked to get the number of pages he still needs to read, the first sentence on our problem shows us that there are 260 pages in the book. Hence, we need to subtract the number of pages Steven has read from the total number of pages of the book. Thus,

260 – 75 = 185

Therefore, Steven has 185 pages left to read.

Multi-Step Word Problem #2

Jesy bought a dozen of boxes, each containing 24 highlighter pens inside. Each box costs \$8. Jesy repacked five of these boxes into packages of six highlighters each and sold them for \$3 per package. She sold the rest of the highlighters at the price of three pens for \$2. How much profit did Jesy make?

• The statement, “Jesy bought a dozen of boxes , each containing 24 highlighter pens inside,” tells us that there are a dozen of boxes that contains 24 highlighters. A dozen means there are 12 boxes . 
• The second sentence, “Each box costs \$8”, means Jesy bought 12 boxes at \$8 each . 
• “Jesy repacked five of these boxes into packages of six highlighters each and sold them for \ $3 per package ” means that Jesy separated 5 boxes from the original 12 boxes to be repacked at a package of six, which was sold at \$3 each. 
• “She sold the rest of the highlighters at the price of three pens for \$2 ” means that Jesy sold the remaining highlighters and bundled it for 3 pens for \$2.

Step 2: Analyze the question of the problem and find the keyword. The last sentence of the problem, “How much profit did Jesy make?” asks us how much profit Jesy earned after repacking the highlighter pens. Profit is defined as the amount earned minus the amount spent to buy the highlighters. 

Step 3: Based on the first statement, Jesy bought 12 boxes containing 24 highlighters. The follow-up statement that says, “Each box costs \$8” refers to the price of each box. In this particular statement, we can find the total expenditures of Jesy for the highlighter pens by simply multiplying the total number of boxes to \$8. Hence, 

12 x \$8 = \$96

This means that Jesy spent \$96 to buy all the highlighters. However, that is not the question being asked. Hence, we need to work on the follow-up statements and find more clues to get Jesy’s profit in selling highlighters. 

Step 4: The next statement says that “Jesy repacked five of these boxes into packages of six highlighters each and sold them for \$3 per package” means that Jesy separated 5 boxes from the dozen to be repacked at a package of six, which was sold at \$3 each. Based on this statement, we need to do three things:

• Find the total number of highlighters she got from separating 5 boxes;
• Find the total number of packages she made by repacking it by 6; and
• Find how much money she made by selling the sets of 6 at \$3.  

Step 5: To find the total number of highlighters she got from separating 5 boxes, we simply multiply 5 by the number of highlighters inside the box. Based on the first statement, each box contains 24 highlighters. Hence,

5 x 24 = 120

This means Jesy repacked a total of 120 highlighter pens.

Step 6: To find the total number of packages she made by repacking 120 highlighter pens by 6, we will divide 120 by 6. Thus,

So, she was able to make 20 sets of 6 highlighter pens. 

Step 7: The next thing we need to do is find how much money she made by selling the sets of 6 by \$3. This can be done by multiplying 20 sets by \$3. Hence,

20 x \$3 = \$60

Thus, Jesy made \$60 from the 20 sets of 6 highlighter pens.

Step 8: The third sentence, “She sold the rest of the highlighters at the price of three pens for \$2” means that Jesy sold the remaining highlighters and bundled it for 3 pens for \$2. From this statement, we need to work on four things first:

• Find the remaining number of boxes; 
• Find the total number of highlighter pens she repacked;
• Find the number of sets she repacked by making sets of 3; and
• Find how much money Jesy made by selling packs of 3 at \$2. 

Step 9: To find the remaining number of boxes, we need to go through some of the problem statements. Based on the first statement, we have 12 boxes, then 5 boxes were separated to make a highlighter set of 6. Hence, we will subtract 5 from 12. 

So, we still have 7 remaining boxes.

Step 10: To find the total number of highlighter pens she repacked, we need to multiply the remaining 7 boxes to the number of highlighter pens inside the box. Going back to the information we already have, we know that there are 24 highlighter pens inside a box. Thus, 

7 x 24 = 168

This means Jesy repacked a total of 168 highlighter pens. 

Step 11: Find the number of sets Jesy made by repacking 168 highlighter pens by 3. This can be done by dividing 168 by 3. Hence,

Thus, Jesy was able to make 56 sets of 3 highlighter pens. 

Step 12: Determine how much money Jesy made by selling each set for \$2. Hence, 

56 x \$2 = \$112

This means Jesy made \$112 by selling 3 highlighter pens for \$2. 

Step 13: The question asks us to determine the profit Jesy made by selling the highlighter pens. In order to find the profit, we need the information of:

• How much did Jesy spend on the highlighter. In Step 3, we found out that she paid \$96 on buying all the highlighter pens. 
• How much money does Jesy make on selling packs of 6 highlighters for \$3. In Step 7, we already know that she made \$60; and
• How much money does Jesy make on selling sets of 3 highlighter pens for \$2. In Step 12, we found out that she made \$112.

Step 14: Before getting the profit Jesy made, we need to know the total money Jesy made in selling the highlighters. Hence, we will simply add the money of \$60 and \$112. Thus,

\$60 + \$112 = \$172

However, \$172 is not the profit Jesy made. This is just the money she was able to make in selling the highlighter pens.

Step 15: Lastly, we will subtract the money Jesy spent on buying the highlighters from the money she made by selling it to find the profit. Thus, 

\ $172 – \$96 = \$76

Therefore, Jesy made a profit of \$76 by selling the highlighter pens.

You can tell that there are lots of things to remember with a multi-step word problem, even when the problem itself is relatively easy. But that’s what makes these problems challenging: you get to use both sides of your brain – your logical math skills and your verbal language skills. That’s why they are often more fun to do than problems that are just numbers without the details and context that word problems give you. The better you understand how to solve them, the more fun they are to solve. 

What are the common errors in solving multi-step word problems?

Mathematical word problems can be challenging to solve. To obtain the correct answer, children must read the words and carefully analyze the problem, determine the appropriate math operation, and then perform the calculations correctly. An error in working on one of the steps may result in a wrong answer. 

Here’s a list of some errors students make when solving multi-step word problems:

• The most common error of students is stopping at one process if they solve one problem. Consider the same word problem about Steven. 

“Steven is reading a book that has 260 pages. He read 35 pages on Monday night and 40 pages on Tuesday night. How many pages does he has left to read?” 

Most students recognize that they need to add 35 and 40 together to get the total number of pages Steven has read so far. Most errors occur when students stop at one process. Adding 35 + 40 will tell you that Steven has read 75 pages so far, but if we go back to the question you are being asked, you will notice that 75 pages are not the answer you are being asked. Thus, we still need to take another step to get there. Steven has read 75 pages so far, but the questions asked us to solve the number of pages he has left to read. Hence, subtracting 75 from the total number of pages of the book makes much more sense. 

• Students get confused with the mathematical operation to use. Even if children are strong readers, they may struggle to pick up on clues in word problems. These clues are phrases that instruct children on how to solve a problem, such as adding or subtracting. The children are then required to convert these phrases into a number sentence in order to solve word problems.

How to teach multi-step problems to children?

There are certain activities or practices that you can try with your child in order to develop their skills in solving multi-step problems. 

• The first and most important skill in working with multi-step is being able to understand the problem clearly. Hence, practicing your child in slowly reading and visualizing problems is the first step in implementing our effective reading comprehension strategies.
• Practice your child in recognizing mathematics terms and vocabulary that are used in word problems. There are keywords or clues that we can easily spot in a word problem if we familiarize ourselves with these mathematical terms. 

Let’s look at the sample words related to addition, subtraction , multiplication, and division.

However, some English words can sometimes be confusing as they may mean differently depending on the context. 

Let’s look at the table below:

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Multiple-Step Word Problems

These multiple-step word problems require students to use reasoning and critical thinking skills to determine how each problem can be solved.

Basic (Grades 2 - 3)

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Intermediate (Grades 3 - 4)

Advanced (grades 4 - 5).

These word problems are sorted by type: addition, subtraction, multiplication, division, fractions and more.

These worksheets, sorted by grade level, cover a mix of skills from the curriculum.

S.T.W. has thousands of worksheets. Visit the full math index to find them all, sorted by topic.

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