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How to Solve Multi-Step Word Problems

Multi-step word problems may initially seem daunting, but with a structured approach, they become manageable and less intimidating. Here, we provide a step-by-step guide to help you navigate these complex problems with ease.

A Step-by-step Guide to Solving Multi-Step Word Problems

Step 1: understand the problem.

The first step in solving multi-step word problems is to read the problem carefully. Look for keywords and phrases that suggest what arithmetic operation(s) you will need to apply. Words like ‘in total’, ‘altogether’ or ‘sum’ suggest an addition, ‘less than’ or ‘remain’ hint towards subtraction, ‘product’ or ‘times’ indicate multiplication, and ‘quotient’ or ‘divided by’ point to division.

Step 2: Identify the Steps Needed

After understanding the problem, list out the necessary steps to reach the solution. Each word problem is a unique puzzle with its sequence of operations. Some problems may require you to perform multiplication before addition, while others may need subtraction followed by division.

Step 3: Assign Variables

For problems with unknown quantities, assign a variable (for example, \(X\) or \(Y\)) to each unknown. This strategy makes it easier to organize information and apply arithmetic operations.

Step 4: Write Equations

Formulate equations based on the identified steps and assigned variables. Keep in mind the order of operations (BIDMAS/BODMAS) – Brackets, Indices/Orders, Division and Multiplication (from left to right), Addition, and Subtraction (from left to right).

Step 5: Solve the Equations

Solving the equations might require simple substitution or more advanced techniques like elimination or matrix method in the case of multiple variables. Don’t forget to check your solutions to make sure they satisfy the original equations.

Step 6: Answer the Question

Finally, ensure that your answer responds to the question asked in the problem. For example, if the problem is asking for the total number of apples, your answer should be a number and mention ‘apples’.

Practical Example

Let’s apply these steps to a sample problem: “Sarah bought \(2\) books. Each book cost twice as much as a pen. She bought \(4\) pens. If each pen cost \($5\), how much did she spend in total?”

Step 1: The problem involves multiplication (each book cost twice as much as a pen) and addition (total amount spent).

Step 2: First, find the cost of a book and then calculate the total cost.

Step 3: Let’s say \(X\) is the cost of a book.

Step 4: The equations will be \(X = 2 \times the\:cost\:of\:a\:pen\) and Total cost = cost of books + cost of pens.

Step 5: Substituting the given cost of a pen (\($5\)), we find \(X = $10\). The total cost is then calculated as \((2 \times $10) + (4 \times $5) = $40\).

Step 6: The total amount Sarah spent is \($40\).

In conclusion, with a systematic approach, you can effectively solve any multi-step word problem. Remember, practice is the key. The more problems you solve, the better you will become at identifying the necessary steps and solving them accurately.

by: Effortless Math Team about 11 months ago (category: Articles )

Effortless Math Team

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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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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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Multi-Step Equations

How to solve multi-step equations.

The word “multi” means more than two, or many. That’s why solving multi-step equations are more involved than one-step and two-step equations because they require more steps.

The main goal in solving multi-step equations, just like in one-step and two-step equations, is to isolate the unknown variable on one side of the equation while keeping the constant or number on the opposite side.

However, there is no rule on where to keep the variable. It all depends on your preference. The “standard” or usual way is to have it on the left side. But there are cases when it makes more sense to keep it on the right side of the equation.

Since we are dealing with equations, we need to keep in mind that whatever operation we perform on one side must also be applied on the other to keep the equation “balanced”.

This concept of performing the same operation on both sides applies to the four arithmetic operations, namely: addition, subtraction, multiplication, and division. For example, if we add 5 on the left side of the equation we must also add 5 on the right side.

Key steps to remember:

• Get rid of any grouping symbols such as square brackets, parentheses, etc, by applying the Distributive Property of Multiplication over Addition.
• Simplify both sides of the equation, if possible, by combining like terms.
• Decide where you want to keep the variable because that will help you decide where to place the constant.
• Eliminate numbers or variables by applying opposite operations: addition and subtraction are opposite operations as in the case of multiplication and division.

Examples of How to Solve Multi-Step Equations

Example 1: Solve the multi-step equation below.

This is a typical problem in multi-step equations where there are variables on both sides. Notice that there is no parenthesis in this equation and no like terms to combine on both sides of the equation.

Clearly, our first step is to decide where to keep or isolate the unknown variable [latex]x[/latex]. Since [latex]7x[/latex] is “larger” than [latex]2x[/latex], then we might as well keep it to the left side. This means we will have to get rid of the [latex]2x[/latex] on the right side. To do that, we need to subtract both sides of the equation by [latex]2x [/latex] because the opposite of [latex]+2x[/latex] is [latex]-2x[/latex].

After doing so, it is nice to see just the variable x on the left side. This implies that we have to move all the constants to the right side by eliminating [latex]+3[/latex] on the left side. The opposite of [latex]+3[/latex] is [latex]-3[/latex], therefore, we will subtract both sides by [latex]3[/latex].

The last step is to isolate the variable [latex]x[/latex] by itself on the left side of the equation. Since [latex]+5[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+5[/latex]. So, we are going to divide both sides by [latex]5[/latex], and then we are done!

Example 2: Solve the multi-step equation below.

Our very first step should be to get rid of the parenthesis by applying the Distributive Property of Multiplication over Addition. That is, multiply [latex] -2[/latex] inside each term in the parenthesis [latex](5-4x)[/latex].

Now, it is time to decide where to keep the unknown variable [latex]x[/latex]. If you decide to keep the variable on the left side, that is perfectly fine.

However, for practice, let’s try keeping it on the right side. We should arrive at the same answers.

To get rid of the [latex]-3x[/latex] on the left side, we add both sides by [latex]3x[/latex] since the opposite of [latex]-3x[/latex] is [latex]+3x[/latex]. After we simplify by adding both sides by [latex]3x[/latex], we obtain this less messy equation.

It’s nice to see the variable x just on the right side. So, we have to move all the constants to the left side.

Clearly, the [latex]-10[/latex] on the right must be removed. The opposite of [latex]-10[/latex] is [latex]+10[/latex], therefore, we will add both sides by [latex]10[/latex]. The last step is to isolate the variable [latex]x[/latex] by itself on the right side of the equation.

Since [latex]+11[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+11[/latex]. So, we are going to divide both sides by [latex]11[/latex] and we are done!

Example 3: Solve the multi-step equation below.

Our first step should be to eliminate the parentheses on BOTH sides of the equation by applying the Distributive property. For the left side, multiply [latex]-4[/latex] inside each term of the parenthesis [latex](4x-8)[/latex] and for the right side, multiply [latex]+3[/latex] inside the parenthesis [latex](-8x-1)[/latex].

Now, before we even decide which side of the equation to isolate the variable, it looks like we have to perform some house cleaning. We need to combine like terms (the x’s ) on the left side of the equation.

Again it doesn’t matter which side to isolate the variable being solved. Say, we decided to keep it on the left.

That means we have to get rid of the [latex]-24x[/latex] on the right side. The opposite of [latex]-24x[/latex] is [latex]+24x[/latex] so we are going to add both sides by [latex]24x[/latex].

Next, we have to move all the constants to the right side of the equation. That [latex]+32[/latex] on the left side must go! The opposite of [latex]+32[/latex] is [latex]-32[/latex], so then we will subtract both sides by [latex]32[/latex].

The last step is to isolate the variable [latex]x[/latex] by itself on the left side of the equation. Since [latex]+5[/latex] is multiplying [latex]x[/latex], then its opposite operation is to divide by [latex]+5[/latex]. And so, let’s divide both sides by [latex]5[/latex] and we are done!

Example 4: Solve the equation [latex]13x – 9x + 20 = 30 + 2[/latex].

Step-by-Step Solution:

1) Combine the variables on the left side of the equation. That is, [latex]13x – 9x=4x[/latex]. Also, simplify the constants on the right side which gives us [latex]30+2=32[/latex].

2) Get rid of [latex]20[/latex] on the left side by subtracting [latex]20[/latex] on both sides of the equation.

3) To solve for [latex]x[/latex], divide both sides by [latex]4[/latex] to get [latex]x=3[/latex].

Example 5: Solve the equation below.

1) Combine like terms on both sides.

2) Subtract [latex]6y[/latex] on both sides to keep the variable [latex]y[/latex] to the left side only.

3) Add [latex]11[/latex] to both sides of the equation.

4) Finally, divide both sides by [latex]-10[/latex] to get the solution.

Example 6: Solve the equation below.

1) Combine the similar terms with variable [latex]m[/latex], and constants on both sides of the equation.

2) Add [latex]5m[/latex] to both sides of the equation. It will keep the variable on the left side and eliminate the variable on the right.

3) Add [latex]14[/latex] to both sides.

4) The last step is to divide the one-step equation by [latex]-3[/latex] to get the value of [latex]m[/latex].

Example 7: Solve the equation [latex]2\left( {x – 5} \right) = 5x + 23[/latex].

1) Eliminate the parenthesis on the left side of the equation by distributing the number outside the parenthesis into the binomial inside.

2) This time, for convenience, we will keep the variable to the right side. To do that, we subtract [latex]2x[/latex] on both sides of the equation.

3) Next, subtract [latex]23[/latex] to both sides of the equation.

4) This leaves to simply divide both sides by the coefficient of [latex]3x[/latex] which is [latex]3[/latex] to get the value of [latex]x[/latex].

Example 8: Solve the equation below.

1) Remove the two parentheses on both sides of the linear equation by applying the Distributive Property of Multiplication over Addition.

2) Combine the constants on both sides. This will clean up the equation tremendously.

3) Add [latex]7h[/latex] to both sides to keep the term with a variable on the left while eliminating the one on the right.

4) Add [latex]57[/latex] on both sides of the equation to keep the constant on the right.

5) Divide both sides by [latex]22[/latex] to get the final solution. That’s it!

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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.

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Multistep Word Problem Examples

Robert had 16 marbles . His brother gave him 3 bags of marbles . If each bag contained 5 marbles , how many marbles does Robert have now?

What's 3 bags × 5 marbles per bag?

3 x 5 = 15!

16 + 15 = 31 marbles

16 + (3 × 5) = ?

16 + ( 3 × 5 ) = 16 + 15 = 31 ✅

Pages to Read in a Book

Sylvia needed to read a book that has 120 pages . She read 26 pages on Friday night, 25 pages on Saturday night, and 18 pages on Sunday night. How many pages did she have left to read?

26 + 25 + 18 = 69 pages !

120 pages - 69 pages = 51 pages

120 - ( 26 + 25 + 18 ) = ?

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Mastering Multi-Step Word Problems

Of all the tricky word problems, multi-step problems cause my students the most trouble. They get confused about the order of events, the operations needed to solve, and the computation of multiple numbers.  In this post, I’m sharing the four simple steps for helping students master multi-step word problems.

Step 1: Read and Visualize

The first thing we do with a multi-step word problem is read it slowly and visualize what is happening. This first step is about using our good reading comprehension strategies.

I ask students to close their eyes and imagine the events as they take place. 

Julie had 82 stickers. 

Imagine a girl with some stickers. Maybe they’re all in a box. See her holding the box in her hands.

Her aunt gave her 26 more stickers.

Can you see her aunt handing her more stickers? Julie adds those to her box now. What operation would match that action?

Then Julie let her brother pick out 12 stickers to keep.

Imagine her little brother coming over and taking 12 out of the box. Will Julie have more or fewer stickers now? What operation matches that step?

How many stickers does Julie have?

Imagine her holding the box of stickers. How many are in there now?

We practice this type of visualizing over and over again until students begin to naturally visualize every time they read a word problem. This is SO important!   Especially with multi-step word problems, you can’t just grab numbers and start working with them without carefully thinking through the operations needed.

During this step, you might also consider leaving out the numbers. You can say, “Julie had some stickers. Her aunt gave her some more. Her brother picked out some to keep.” This can help students focus on the events rather than trying to solve!

Step 1 (Part 2): Act it Out

I like to have students visualize first so that each child has to do the thinking work on their own.  Visualizing is an important strategy for independent problem solving.

But when we’re working together as a whole class, it’s helpful to make sure everyone is on the same page and was able to visualize the situation correctly.  That’s why we also physically act out multi-step word problems.

We act out the problems with simple gestures that show if we are giving away (losing, eating, using up, etc.) things or if we are gaining more of something.  Incorporating movement helps many students make sense of tricky problems.

Sometimes, I’ll ask for a few volunteers to act out the problem in front of the classroom, and sometimes we’ll all just do it together from our spots. 

Again, it’s often helpful to leave out the numbers. Our goal is to give meaning to what is happening in the problem and relate each action to its matching operation, not to start solving.

Step 2: Write an Answer Statement First

Once students have made sense of the problem, it’s helpful to have them write an Answer Statement.  The Answer Statement can help students determine and remember what the problem is asking them to find out.

For example:

Julie has ___ stickers now.

This helps students remember that we are trying to determine how many stickers Julie has after the events of the problem. It also ensures that students actually READ the question!

After students solve the problem, they can place their answer in the statement and check if it makes sense.

Step 3: Write the Full Equation

One of the challenges of multi-step word problems is that students often solve one part but forget or miss the next step.  I find that it’s helpful to have them write the entire equation, including all the steps, before trying to solve any of the parts.

From the earlier example, students would write:

82 + 26 -12 = ?

After they have written the equation, I like to look back at it and talk out how it matches the story problem.

Julie started with 82 stickers. Her aunt gave her 26 more, so we need to add 26. Then her brother took 12, so we need to subtract 12.

Now students can see all of the events represented in one equation and will be less likely to forget a step.

Step 4: Chunk the Parts

Now that students know the parts, I ask them to look at the equation and determine a plan for solving. In many cases, this simply means to complete the first part (86 + 26 = A) and then use that answer to complete the second part. (A – 12 = ?)

In some cases, students may notice an easier way to work with the numbers and reorder the steps.

125 + 52 – 25 = ?

Some students may see that they can simply take the 25 from 125 first and end up with 100, which is then easy to add to 52.

Another example:

42 – 18 + 7 = ?

Some students may realize they can avoid regrouping by first adding 7 to 42 and then subtracting the 18. 

When students know all the steps, they are able to think about smarter ways to complete the task.  

Handy Tools for Multi-Step Word Problems

I find that the key to teaching multi-step word problems is starting with a simplified version of the problem and working up to the more challenging ones.

My favorite way to do this is to use Tiered Word Problems . These are leveled word problems that slowly increase in challenge from the first grade level on up.

Since multi-step word problems can be scary, I decided to use a zombie theme for the pack. (Plus, my students love zombies!) This pack includes leveled problems that allow me to differentiate instruction and gradually increase the challenge. 

Although the pack is targeted to 2nd and 3rd graders, it’s perfect for challenging younger students or intervening when older students aren’t quite catching on. 

You can check out the Multi-Step Word Problems Zombie Pack in my TPT store.

I hope these tips make your teaching life a little easier! What strategies help your students make sense of multi-step problems? Comment below!

You might also like:

Word Problems Made Easy

Problem Solving Rounds for Teaching Word Problems

Guided Math Groups for Teaching Word Problems

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• Solving Multi-Step Equations – Methods & Examples

Solving Multi-Step Equations – Methods & Examples

One step equation is an equation that requires only one step to be solved.  You only perform a single operation in order to solve or isolate a variable. Examples of one step equations include: 5 + x = 12, x – 3 = 10, 4 + x = -10 etc.

• For instance, to solve 5 + x = 12,

You only need to subtract 5 from both sides of the equation:

5 + x = 12 => 5 – 5 + x = 12 – 5

=> x = 7

To solve this equation, divide both sides of the equation by 3.

You can note that for a one-step equation to be completely solved, you only need a single step: add/subtract or multiply/divide.

A two-step equation, on the other hand, requires two operations to perform to solve or isolate a variable. In this case, the operations to solve a two-step are addition or subtraction and multiplication or division. Examples of two-step equations are:

• (x/5) – 6 = -8

Add both 6 to both sides of the equation and multiply by 5.

(x/5) – 6 + 6 = – 8 + 6

(x/5)5 = – 2 x 5

• 3y – 2 = 13

Add 2 to both sides of the equation and divide by 3.

3y – 2 + 2 = 13 + 2

3y/3 = 15/3

• 3x + 4 = 16.

To solve this equation, subtract 4 from both sides of the equation,

3x + 4 – 4 = 16 – 4.

This gives you the one-step equation 3x = 12. Divide both sides of the equation by 3,

3x/3 = 12/3

What is a Multi-step Equation?

The term “multi” means many or more than two. Therefore, a multi-step equation can be defined as an algebraic expression that requires several operations such as addition, subtraction, division, and exponentiation to be solved. Multi-step equations are solved by applying similar techniques used in solving one-step and two-step equations.

Like we saw in one-step and two-step equations, the main objective of solving multi-step equations is to isolate the unknown variable on either the RHS or LHS of the equation while keeping a constant term on the opposite side. The strategy of obtaining a variable with a coefficient of one entails several processes.

The Law of equations is the most important rule you should remember while solving any linear equation. This implies that, whatever you do to one side of the equation, you MUST do to the opposite of the equation.

For instance, if you add or subtract a number on one side of the equation, you must also add or subtract on the equation’s opposite side.

How to Solve Multi-step Equations?

A variable in an equation can be isolated on any side, depending on your preference. However, keeping a variable on the left side of the equation makes more sense because an equation is always read from left to right.

When solving algebraic expressions , you should keep in mind that a variable doesn’t need to be x. Algebraic equations make use of any available alphabetical letter.

In summary, to solve multi-step equations, the following procedures are to be followed:

• Eliminate any grouping symbols such as parentheses, braces, and brackets by employing the distributive property of multiplication over addition.
• Simplify both sides of the equation by combining like terms.
• Isolate a variable on any side of the equation depending on your preference.

Examples of How to Solve Multi-Step Equations

Solve the multi-step equation below.

12x + 3 = 4x + 15

This is a typical multi-step equation where variables are on both sides. This equation has no grouping symbol and like terms to combine on opposite sides. Now, to solve this equation, first decide where to keep the variable. Since 12x on the left side is greater than 4x on the right side, therefore we keep our variable to the LHS of the equation.

This implies that, we subtract by 4x from both sides of the equation

12x – 4x + 3 = 4x – 4x + 15

6x + 3 = 15

Also subtract both sides by 3.

6x + 3 – 3 = 15 – 3

The last step now is to isolate x by dividing both sides by 6.

6x/6 = 12/6

And there, we are done!

Solve for x in the multi-step equation below.

-3x – 32 = -2(5 – 4x)

• The first step is to remove the parenthesis by use of the Distributive Property of Multiplication.

-3x – 32 = -2(5 – 4x) = -3x – 32 = – 10 + 8x

• In this example, we have decided keep the variable on the left side.
• adding both sides by 3x gives; -3x + 3x – 32 = – 10 + 8x + 3x =>

– 10 + 11x = -32

• Add both sides of the equation by 10 to clear -10.

– 10 + 10 + 11x = -32 + 10

• Isolate the variable x by dividing both sides of the equation by 11.

11x/11 = -22/11

Solve the multi-steps equation 2(y −5) = 4y + 30.

• Remove the parentheses by distributing the number outside.

= 2y -10 = 4y + 30

• By keeping the variable to the right side, subtract 2y from both sides of the equation.

2y – 2y – 10 = 4y – 2y + 23

-10 = 2y + 30

• Next, subtract both of the sides of the equation by 30.

-10 – 30 = 2y + 30 – 30

– 40 = 2y

• Now divide both sides by the coefficient of 2y to get the value of y.

-40/2 = 2y/2

Solve the multi-steps equation below.

8x -12x -9 = 10x – 4x + 31

• Simplify the equation by combining like terms on both sides.

– 4x – 9 = 6x +31

• Subtract on both sides of the equation by 6x to keep the variable x to the equation’s left side.

– 4x -6x – 9= 6x -6x + 31

-10x – 9 = 31

• Add 9 to both sides of the equation.

– 10x -9 + 9 = 31 +9

• Finally, divide both sides by -10 to get the solution.

-10x/-10 = 40/-10

x = – 4

Solve for x in the multi-step equation 10x – 6x + 17 = 27 – 9

Combine the like terms on both sides of the equation

4x + 17 = 18

Subtract 17 from both sides.

4x + 17 – 17 = 18 -17

Isolate x by dividing both sides by 4.

-3x – 4(4x – 8) = 3(- 8x – 1)

The first step is to remove the parentheses by multiplying numbers outside the parentheses by terms within the parentheses.

-3x -16x + 32 = -24x – 3

Perform a bit of housecleaning by collecting like terms on both sides of the equation.

-19x + 32 = -24x – 3

Let’s keep our variable to the left by adding 24x to both sides of the equation.

-19 + 24x + 32 = -24x + 24x – 3

5x + 32 = 3

Now move all constants to right side by subtracting by 32.

5x + 32 -32 = -3 -32

The last step is to divide both sides of the equation by 5 to isolate x.

5x/5 = – 35/5

Solve for t in the multi steps equation below.

4(2t – 10) – 10 = 11 – 8(t/2 – 6)

Apply the distributive property of multiplication to eliminate the parentheses.

8t -40 – 10 = 11 -4t – 48

Combine the like terms on both sides of the equation.

8t -50 = -37 – 4t

Let’s keep the variable on the left side by adding 4t to both sides of the equation.

8t + 4t – 50 = -37 – 4t + 4t

12t – 50 = -37

Now add 50 to both sides of the equation.

12t – 50 + 50 = – 37 + 50

Divide both sides by 12 to isolate t.

12t/12 = 13/12

Solve for w in the following multi steps equation.

-12w -5 -9 + 4w = 8w – 13w + 15 – 8

Combine the like term and constants of both sides of the equation.

-8w – 14= -5w + 7

To keep the variable on the left side, we add 5w on both sides.

-8w + 5w – 14 = -5w + 5w + 7

-3w – 14 = 7

Now add 14 to both sides of the equation.

– 3w – 14 + 14 = 7 + 14

The final step is to divide both sides of the equation by -3

-3w/-3 = 21/3

Practice Questions

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Multi Step Equations

Multi step equations are equations that need more than two steps to solve for the variable. We use the same operation on both sides of the equation to such equations. Solving multiple step equations is sometimes complicated when compared to one step or two-step equations as they require multiple steps.

Let us see how to solve multi-step equations and the properties that we use for the same.

What are Multi Step Equations?

Multi step equations are equations that require more than one operation (applied on both sides) to solve for the required variable. Often word problems also may lead to multi step equations. They look complicated . Here are some examples of multi step equations:

• 2 (4x - 5) = 3x - 7
• –3 [(1/5)t + 1/3] = 9
• 16 - (2x + 1) = (5x - 2) + 13

We already know how to solve one-step and two-step equations . We just extend the same process to solve the multi step equations as well. i.e., we use the properties of equations (like adding, subtracting, multiplying, or dividing both sides by some number/variable such that the equation is balanced) to solve the multi step equations. Let us see how to solve them.

Inverse Operations For Solving Multi-Step Equations

We solve the multi step equations by applying inverse operation on both sides to isolate the variable (making the variable alone on one side of the equation). Note that equality should not be disturbed when we apply any operation. For this, we should apply the same operation on both sides. For example, to solve x + 2 = 3 (of course, this is not a multi step equation), we should subtract 2 from both sides, then we get x + 2 - 2 = 3 - 2 which gives x = 1. Here since we are subtracting 2 from the left side, we should subtract the same number 2 from the right side as well. Wait! Why did we "subtract" 2? Because in the original equation x + 2 = 3, 2 was getting added to x, and to solve for x, we do NOT need + 2 on the left side, so we have subtracted it (as subtraction is the inverse operation of addition). Let us quickly revise the inverse operations:

• Addition and subtraction are inverse operations of each other
• Multiplication and division are inverse operations of each other
• Exponents and roots are inverse operations of each other Example: square and square root are inverse operations of each other, cube and cube root are inverse operations of each other, etc).

Applying the same operation on both sides without affecting the equality is proposed by the properties of equations. Here are some examples to understand them.

Solving Multi Step Equations

To solve multi step equations we may need to apply multiple types of inverse operations one after the other (that are mentioned in the previous section). The order of applying inverse operations is very important while solving multi step equations. For example, to solve the equation 2x + 4 = 6, the first step is NOT dividing both sides by 2, rather we subtract 4 from both sides. i.e.,

2x + 4 = 6 Subtracting 4 from both sides, 2x = 2 Dividing both sides by 2, x = 1

Our ultimate aim is to get just the variable on one side of the equation. We should aim at getting the answer something like "variable = something". Here are the important steps to solve multi step equations:

• Apply distributive property when you have a parenthesis.
• Combine like terms (if any).
• Collect like terms to one side of the equation. i.e., collect variable terms on the left side and the constants on the right side (or vice versa).
• Isolate the variable by inverse operations.

Here is an example to understand these steps.

Example: -2 (x - 3) - 7 = 7x + 11

This is a multi step equation with variable on both sides.

Applying distributive property (i.e., distributing -2 to the terms inside the brackets),

-2x + 6 - 7 = 7x + 11

Combing like terms (i.e., 6 - 7 = -1),

-2x - 1 = 7x + 11

Now our aim is to collect all x terms on the left and all constant on the right.

Subtracting 7x from both sides,

-9x - 1 = 11

Adding 1 on both sides,

Our aim is fulfilled now. Now, let us divide both side by -9,

Since x is isolated, it means that we have solved the equation.

Multi Step Equations with Fractions

Sometimes, multi step equations may involve one or more fractions in them. The easiest way of solving such equations is

• Find the LCD (Least Common Denominator) of all the denominators (of both left and right sides).
• Multiply every term on both sides of the equation by LCD .
• Apply inverse operations and isolate the variable.

Here is an example.

Example: Solve (1/4)t + 1/5 = (1/2)t + 5/3

The denominators are 4, 5, 2, and 3. Their LCD is 60. So multiply each term on both sides by 60.

(1/4)t × 60 + 1/5 × 60 = (1/2)t × 60 + 5/3 × 60

15t + 12 = 30t + 100

Now, the equation is free from fractions. We will proceed now. Subtracting 30t and 12 from both sides,

Dividing both sides by -15,

t = -88/15.

☛ Related Topics:

• Multi-Step Equation Calculator
• System of Equations Calculator
• Equation Calculator
• Two-Step Equations Calculator

Multi Step Equations Examples

Example 1: Solve the equation for "s": 3 (4s + 1) = 9.

We can solve the given multi step equation in two ways.

3 (4s + 1) = 9 Dividing both sides by 3, 4s + 1 = 3 Subtracting 1 from both sides, 4s = 2 Dividing both sides by 4, s = 1/2

3 (4s + 1) = 9 Distributing 3, 12s + 3 = 9 Subtracting 3 from both sides, 12s = 6 Dividing both sides by 12, s = 1/2

Answer: The solution is s = 1/2.

Example 2: Solve (1/3) x + 5 = 6x for x.

The given equation is:

(1/3) x + 5 = 6x

Multiply each term on both sides by 3 to avoid the fraction. x + 15 = 18x Let us collect the variables on one side. Subtracting x from both sides, 15 = 17x Dividing both sides by 17, x = 15/17

Answer: The solution of the given multi-step equation is x = 15/17.

Example 3: John is 5 years elder than his brother Michael. After 10 years, the sum of their ages is 35. Then how old is Michael now?

This is a multi-step equations word problem.

Let Michael is x years old. Then John's age = x + 5. After 10 years, the sum of their ages is 35. i.e.,

(x + 10) + (x + 5 + 10) = 35

Combining like terms ,

2x + 25 = 35

Subtracting 25 from both sides,

Dividing both sides by 2,

Answer: Michael is 5 years old.

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FAQs on Multi Step Equations

How do you solve multi step equations.

To solve multi step equations , aim at getting the required variable on the left side of the equation. All the other numbers/variables should be removed from the left side by using inverse operations. The inverse operation of addition is subtraction (and vice versa) and the inverse operation of multiplication is division (and vice versa).

How to Solve Multistep Equations With Variables on Both Sides?

If a multi step equation has variables on both sides, then apply inverse operations to get the variable terms on one side and constant terms on the other side. Then solve for x. For example: 3x + 5 = 7x + 6 ⇒ 3x + 5 - 7x = 6 ⇒ -4x + 5 = 6 ⇒ -4x = 6 - 5 ⇒ -4x = 1 ⇒ x = -1/4.

How to Solve Multi Step Equations With Fractions?

If a multi step equation has fractions, then we can eliminate all fractions first by multiplying each term on both sides by LCD of all denominators. Then we can just apply the inverse operations and solve for the variable.

What are the 4 Steps of Multi Step Equations?

To solve multi step equations:

• Expand brackets by using distributive property if any.
• Combine like terms if any.
• Collect like terms of one type on either side.
• Apply inverse operations to isolate the variable.

Where to Find Multistep Equations Worksheets?

We can find multi step equations worksheets by clicking here . You can get varieties of problems (both equations and word problems) by clicking on the given link.

How to Solve Multi Step Equations Word Problems?

To solve multi step equations word problems:

• First, identify the variables with respect to the given context.
• Frame the multi step equation by reading the problem carefully.
• Solve it using the method that is explained on this page.

The multi step equation can also be solved directly through graph and it is often used in linear programming .

Solving Multi Step Equations – Definition, Facts, Examples, FAQs

What are multi-step equations, inverse operations used for solving multi-step equations, how to solve multi-step equations, solved examples on multi-step equations, practice problems on multi-step equations, frequently asked questions about multi-step equations.

The multi-step equations are algebraic equations that require multiple steps (more than two steps) to solve.  

Solving multi-step equations in algebra is similar to solving one-step and two-step equations, but the process is a little lengthy as there are multiple steps involved.

Examples of Multi-step Equations:

• $3 (4x \;-\; 5) = 4x \;-\; 7$
• $-\;1\left[(\frac{1}{7})m + \frac{1}{6}\right] = 4$
• $10 \;-\; (2t + 3) = (3t \;-\; 5) + 20$

We are familiar with solving one-step and two-step equations. 

• One-step equations are equations that can be solved in a single step. 
• Two-step equations require only two steps to solve.

Multi-step equations involve more complex operations. For solving the multi-step equations, we just extend the same process. 

Multi-step Equations: Definition

Multi-step equations are the equations in which we have to use multiple steps (to undo more than two operations) to isolate the variable and solve. 

Related Worksheets

To solve the multi-step equations, we need to isolate the variable. We can do that by applying the inverse operation on both sides. To balance the equation, we apply the same operation on both sides. 

Also, powers and roots are the opposite operations which undo each other.

Step 1: Simplify both sides of the equation by expanding the brackets, combining like terms, and using the distributive property if necessary.

Step 2: Isolate the variable term by performing inverse operations on both sides of the equation.

Step 3: Continue simplifying until the variable term is isolated on one side and all other terms are shifted to the other side.

Step 4: Solve for the variable by performing the necessary operations to isolate the variable.

Step 5: Verify your solution by substituting the value back into the original equation.

Example 1: $2x + 5 \;-\; 3(2 \;-\; x) = 4x \;-\; 7$

$2x + 5 \;-\; 3(2 \;-\; x) = 4x \;-\; 7$

$2x + 5 \;-\; 6 + 3x = 4x \;-\; 7$ … expanding the bracket

$2x + 3x \;-\;4x + 5 \;-\; 6 = \;-\; 7$ …combining like terms

$x + 5 \;-\; 6 = \;-\; 7$

$x = 6 \;-\; 5 \;-\; 7$ …isolating the variable

$x = \;-\; 6$

Example 2: $–\; 2(y + 3) = 3(y \;–\; 7) + 5$

$–\; 2(y + 3) = 3(y \;–\; 7) + 5$

$\;–\; 2y \;–\; 6 = 3y \;–\; 21 + 5$ …applying the distributive property 

$–\; 2y \;–\; 6 = 3y \;–\; 21 + 5$

$–\; 2y \;–\; 6 = 3y \;–\; 16$

$–\; 2y \;–\; 3y \;–\; 6 = 3y \;–\; 3y \;–\; 16$ …combining like terms

$–\; 5y \;–\; 6 = \;–\; 16$

$–\; 5y = \;– \;10$ …adding 6 on both sides

$y = \frac{-\;10}{-\; 5}$ …dividing both sides by -5

Solving Multi-step Equations Involving Fractions

If a multi-step equation involves one or more fractions in them, we need to apply the rules for adding or subtracting fractions with same or different denominators when solving the equation. 

To get rid of fractions, use the following steps:

Step 1: Find the Least Common Denominator (LCD) of all the denominators to make the denominators same.

Step 2: Multiply each and every term on both sides of the equation by LCD.

Step 3: Apply inverse operations and isolate the variable.

$\frac{m}{5} + \frac{1}{6} = \frac{1}{3} + \frac{1}{2}m$

Step 1: The denominators are 2, 3, 5, and 6.

LCM of 2, 3, 5, and 6 is 30.

LCD of $\frac{1}{5},\;\frac{1}{6},\;\frac{1}{3}$, and $\frac{1}{2}$ is 30.

Step 2: Multiply each term on both sides by 30.

$30(\frac{m}{5} + \frac{1}{6}) = 30(\frac{1}{3} + \frac{1}{2}m)$

$6m + 5 = 10 + 15m$

Now, the above equation is simplified and free from fractions. 

Subtracting 6m and 10 from both the sides, we get

$6m + 5\;-\;6m\;-\;10 = 10\;-\;10 + 15m\;-\;6m$

$-\;5 = 9m$

Dividing both the sides by 9, we get

$m = \frac{-\;5}{9}$

Multi-step Equations: Word Problems

Sometimes we have to form the equation using the information provided in the word problem.

Example: Find the angle measures of a triangle if the first angle is thirty degrees more than

the second, and the third is ten degrees less than twice the first.

Let the three vertices of a triangle be A, B, and C. We know that the sum of three angles of a triangle is 180 o .

Thus, m∠A + m∠B + m∠C = 180 o …(1)

Suppose that the first angle be ∠A. The first angle is thirty degrees more than the second.

m∠A = m∠B + 30 o

Let m∠B = x

Thus, m∠A = x + 30 o …(2)

The third angle is ten degrees less than twice the first.

m∠C = 2 m∠A – 10 o

But m∠A = x + 30 o

Thus, m∠C = 2 (x + 30 o ) – 10 o

m∠C = 2x + 50 o …(3)

Substitute the values from (2) and (3) in (1).

m∠A + m∠B + m∠C = 180 o

x + 30 + x + 2x + 50 = 180 o

x + x + 2x + 80= 180 o

Thus, m∠B = x = 25 o

 m∠A = x + 30 o = 55 o

 m∠C = 2x + 50 o = 100 o

Facts about Multi-step Equations

• Multi-step equations require a sequence of steps to solve, such as combining like terms, distributing, and applying inverse operations.
• Isolating a variable simply means bringing the variable on one side and all the other

In this article, we learned about multi-step equations, how to solve them, and a few examples to understand the steps involved. Let’s solve a few multi-step equations and practice problems based on multi-step equations for better understanding.

1. Find the solution of $\frac{1}{2}(2x \;-\; 4) = 11$

$\frac{1}{2}(2x\;-\;4) = 11$

Multiply both sides by 2, we get

$(2x\;-\;4) = 11 \times 2$

$2x\;-\;4 = 22$

Adding 4 on both sides, we get

$2x\;-\;4 + 4 = 22 + 4$

Dividing both the sides by 2, we get

$x = \frac{26}{2}$

2. Solve: $\frac{1}{4}x\;-\;\frac{3}{4} = 2x + 3$

$\frac{1}{4}x\;-\;\frac{3}{4} = 2x + 3$

Multiplying both the sides by 4, we get

$x\;-\;3 = 8x + 12$

Subtracting 8x and adding 3 from both the sides, we get

$x\;-\;3\;-\;8x + 3 = 8x + 12\;-\;8x + 3$

$-\;7x = 15$

$x =\frac{-\;15}{7}$

3. Solve the equation for “m”: $4(5s + 3) = 12$

$4(5s + 3) = 12$

Distributing 4 on LHS, we get

$20s + 12 = 12$

Subtracting 12 from both sides, we get

$20s + 12 \;-\; 12 = 12 \;-\; 12$

Dividing both sides by 20, we get

4. Solve $\frac{x}{4} \;-\; 5 = 3x$ for x .

Solution: 

The given equation is:

$\frac{x}{4}\;-\;5 = 3x$

Multiply each term on both sides by 4.

$x\;-\;20 = 12x$

Subtracting x from both sides, we get

$x\;-\;20\;-\;x = 12x\;-\;x$

$-\;20 = 11x$

Dividing both sides by 11, we get

$x = \frac{-\;20}{11}$

5. Mia is 5 years older than Molly. After 5 years, the sum of their ages is 25. Then how old is Mia now?

Suppose that Molly’s current age is x years. 

Mia’s age = x + 5.

Molly’s age after 5 years = x + 5

Mia’s age after 5 years = x + 10

Sum of their ages after 5 years = 25

(x + 5) + (x + 10) = 25

2x + 15 = 25 Combining like terms

2x = 10 Subtracting 25 from both sides

x = 5 Dividing both sides by 2

Molly’s age = 5 years

Mia’s age = 5 + 5 = 10 years

Solving Multi Step Equations - Definition, Facts, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the value of y in $2(y\;-\;10) = 3(y + 5)$?

$\frac{t + 5}{t \;-\; 1} = \frac{3}{2}$. find t., find the value of s. $-\;5s + 4(3s \;-\; 6) = 2(\;-\;6s\;-\;1)$., the digit in tens place is twice the digit in ones place. if the digit at tens place is c, what will be the number, the sum of three consecutive multiples of 9 is 3024. which of the following is the smallest multiple among them.

What is a one-step equation?

One-step equation is an algebraic equation which we can solve in just one step. Example of a one-step equation is x + 4 = 5.

What is a two-steps equation?

Two steps equation is an algebraic equation which we can solve in two steps. Example of a two-step equation is 2y + 5 = 6.

How do I know if an equation is multi-step?

An equation is considered multi-step if it involves more than one arithmetic operation to solve. It generally involves multiple steps, such as combining like terms, distributing, or simplifying expressions before isolating the variable.

What is the order of operations in solving multi-step equations?

The order of operations in solving multi-step equations follows the same rules as in general mathematics, which is PEMDAS. 

Parentheses, Exponents, Multiplication and Division (same level), and Addition and Subtraction (same level), in order from left to right.

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Solving Multi-Step Equations

In these lessons, we will learn how to

• Solve multi-step equations with whole numbers
• Solve multi-step equations with fractions
• Solve multi-step equations with decimals

Related Pages Solving Algebraic Equations More Algebra Lessons

Solving multi-step equations with whole numbers

The following diagram shows how to solve multi-step equations. Scroll down the page for more examples and solutions.

To solve a multi-step equation, we would start by trying to simplify the equation by combining like terms and using the distributive property whenever possible.

Consider the equation 2(x + 1) – x = 5. First, we will use the distributive property to remove the parenthesis and then we can combine like terms and then isolate the variable.

Example: Solve 2(x + 1) – x = 5

Solution: 2(x + 1) – x = 5 2x + 2 – x = 5 (use distributive property) x + 2 = 5 (combine like terms) x + 2 – 2 = 5 – 2 x = 3

How to solve multi-step equations by combining like terms and using the distributive property?

• 4x + 2x - 3x = 27
• 4a + 1 - a = 19
• 4(y - 1) = 36
• 16 = 2(x - 1) - x

Use distributive property to simplify multi-step equations

• 7(w + 20) - w = 5
• 9(x - 2) = 3x + 3
• Lydia inherited a sum of money. She split it into five equal chunks. She invested three parts of the money in a high interest bank account which added 10% to the value. She placed the rest of her inheritance plus $500 in the stock market but lost 20% on that money. If the two accounts end up exactly the same amount of money in them, how much did she inherit?

Solving Multi-Step Equations With Fractions

To solve an equation with fractions, we first try to change it into an equation without fractions. Then, we can solve it using the methods we already know.

How to solve multi-step equations with fractions?

• 1/4 x + 3 = 2
• 1/2 (k - 8) = 6
• 1/2 d + 2 = 3/4
• -3/4 x + 1/4 = 1/2

How to solve Multi-Step Equations with Fractions & Decimals?

• 3/2 n - 6 = 22
• 2/5 x + 2 = 3/4
• 0.035m + 9.95 = 12.75

Solving Multi-Step Equations With Decimals

The steps involved in solving multi-step equations with decimals are the same as those in equations with whole numbers. The complication may lie more in the multiplication and division of decimals rather than the steps. Another method would be to multiply each term of the equation by ten (or hundred) to convert the decimals to whole numbers and then solve the equation.

How to solve multi-step equations with decimals?

• 0.4x + 9.2 = 10
• 0.4(a + 2) = 2
• 1.2c + 2.6c = 4.56

How to solve multiple step linear equations involving decimals?

• Remove parentheses by using the distributive property. Then combine like terms on each side.
• Add or subtract, as needed, to get all variable terms on one side and all constant terms on the other. Then combine like terms.
• Multiply or divide to solve for the variable.
• Check all possible solutions.

Examples: Solve each equation.

• 1.2x - 5.12 - 0.9x = 1.6
• 5x - 0.2(x - 4.2) = 1.8

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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.