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Systems of Equations Word Problems Worksheets

The word problems given on these printable worksheets require you to set up a system of equations and solve for the unknown quantities.

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Word Problems Worksheet 1 – This 6 problem algebra worksheet will help you practice creating and solving systems of equations to represent real-life situations. You will use the “ elimination ” method to eliminate variables from standard form equations. Word Problems Worksheet 1 RTF Word Problems Worksheet 1 PDF View Answers

Word Problems Worksheet 3 – This 6 problem algebra worksheet will help you practice creating and solving systems of equations to represent real-life situations. Most of the problems involve money, so make sure you’re ready for some decimals. Word Problems Worksheet 3 RTF Word Problems Worksheet 3 PDF View Answers

Word Problems Worksheet 4 – This 6 problem algebra worksheet will help you practice creating and solving systems of equations to represent real-life situations. Most of the problems involve money, and a few distractors are introduced. Word Problems Worksheet 4 RTF Word Problems Worksheet 4 PDF View Answers

Word Problems Worksheet 5 – This 8 problem algebra worksheet features more abstract word problems like “ The sum of x and y is 42.  the difference of x and y is 13.  Find x and y .” There are a few negative integers, so be careful! Word Problems Worksheet 5 RTF Word Problems Worksheet 5 PDF View Answers

Word Problems Worksheet 6 – This 8 problem algebra worksheet features more abstract word problems like “ The sum of  twice a number, x, and twice another number, y, is 118.  The value of y is one less than twice the value of x.  Find x and y. ” One of the problems even has an infinite number of solutions! Word Problems Worksheet 6 RTF Word Problems Worksheet 6 PDF View Answers

These free  systems of equations   worksheets  will help you practice solving real-life systems of equations using the “ elimination ” method.  You will  need to create and solve a system of equations to represent each situation.  The exercises can also be solved using other algebraic methods if you choose.

This is a progressive series that starts simple with problems involving buying movie tickets and collecting for fundraisers.  Eventually the negative numbers , decimals , the  distributive property  and “ the opposite of x ” come into play.

Each worksheet will help students master Common Core skills in the Algebra strand.  They are great for ambitious students in pre-algebra or algebra classes.

These free  elmination  worksheets are printable and available in a variety of formats.  Each sheet includes an example to help you get started.  Of course, answer keys are provided as well.

Philip Teas

There are multiple problems with your worksheets. You have word problems which are unanswerable or answers which make no sense such as fractions of people. Your answer sheets are frequently incorrect. For a quick example: System of Equations- Word Problems #1 – KEY. Question #1 states that there are 9 tickets purchased, the Answer shows a=6 and b=2. 6+2 is 8 not 9. Question 5 on that same worksheet is unanswerable. I’ve found problems on worksheet 1, 2, 5 so far. How can you claim to have the best materials with so many clear problems?

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Solving Systems of Equations Real World Problems

Wow! You have learned many different strategies for solving systems of equations! First we started with Graphing Systems of Equations . Then we moved onto solving systems using the Substitution Method . In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations.

Now we are ready to apply these strategies to solve real world problems! Are you ready? First let's look at some guidelines for solving real world problems and then we'll look at a few examples.

Steps For Solving Real World Problems

• Highlight the important information in the problem that will help write two equations.
• Define your variables
• Write two equations
• Use one of the methods for solving systems of equations to solve.
• Check your answers by substituting your ordered pair into the original equations.
• Answer the questions in the real world problems. Always write your answer in complete sentences!

Ok... let's look at a few examples. Follow along with me. (Having a calculator will make it easier for you to follow along.)

Example 1: Systems Word Problems

You are running a concession stand at a basketball game. You are selling hot dogs and sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the night you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined. You must report the number of hot dogs sold and the number of sodas sold. How many hot dogs were sold and how many sodas were sold?

1.  Let's start by identifying the important information:

• hot dogs cost $1.50
• Sodas cost $0.50
• Made a total of $78.50
• Sold 87 hot dogs and sodas combined

2.  Define your variables.

• Ask yourself, "What am I trying to solve for? What don't I know?

In this problem, I don't know how many hot dogs or sodas were sold. So this is what each variable will stand for. (Usually the question at the end will give you this information).

Let x = the number of hot dogs sold

Let y = the number of sodas sold

3. Write two equations.

One equation will be related to the price and one equation will be related to the quantity (or number) of hot dogs and sodas sold.

1.50x + 0.50y = 78.50    (Equation related to cost)

 x + y = 87   (Equation related to the number sold)

4.  Solve! 

We can choose any method that we like to solve the system of equations. I am going to choose the substitution method since I can easily solve the 2nd equation for y.

5. Think about what this solution means.

x is the number of hot dogs and x = 35. That means that 35 hot dogs were sold.

y is the number of sodas and y = 52. That means that 52 sodas were sold.

6.  Write your answer in a complete sentence.

35 hot dogs were sold and 52 sodas were sold.

7.  Check your work by substituting.

1.50x + 0.50y = 78.50

1.50(35) + 0.50(52) = 78.50

52.50 + 26 = 78.50

35 + 52 = 87

Since both equations check properly, we know that our answers are correct!

That wasn't too bad, was it? The hardest part is writing the equations. From there you already know the strategies for solving. Think carefully about what's happening in the problem when trying to write the two equations.

Example 2: Another Word Problem

You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos and your total bill is $11.25. Your friend's bill is $10.00 for four soft tacos and two burritos. How much do soft tacos cost? How much do burritos cost?

• 3 soft tacos + 3 burritos cost $11.25
• 4 soft tacos + 2 burritos cost $10.00

In this problem, I don't know the price of the soft tacos or the price of the burritos.

Let x = the price of 1 soft taco

Let y = the price of 1 burrito

One equation will be related your lunch and one equation will be related to your friend's lunch.

3x + 3y = 11.25  (Equation representing your lunch)

4x + 2y = 10   (Equation representing your friend's lunch)

We can choose any method that we like to solve the system of equations. I am going to choose the combinations method.

5. Think about what the solution means in context of the problem.

x = the price of 1 soft taco and x = 1.25.

That means that 1 soft tacos costs $1.25.

y = the price of 1 burrito and y = 2.5.

That means that 1 burrito costs $2.50.

Yes, I know that word problems can be intimidating, but this is the whole reason why we are learning these skills. You must be able to apply your knowledge!

If you have difficulty with real world problems, you can find more examples and practice problems in the Algebra Class E-course.

Take a look at the questions that other students have submitted:

Problem about the WNBA

Systems problem about ages

Problem about milk consumption in the U.S.

Vans and Buses? How many rode in each?

Telephone Plans problem

Systems problem about hats and scarves

Apples and guavas please!

How much did Alice spend on shoes?

All about stamps

Going to the movies

Small pitchers and large pitchers - how much will they hold?

Chickens and dogs in the farm yard

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How to Solve Systems of Equations Word Problems? (+FREE Worksheet!)

Learn how to create and solve systems of equations word problems by the elimination method.

Related Topics

• How to Solve One-Step Equations
• How to Solve One-Step Inequalities
• How to Solve Multi-Step Inequalities
• How to Solve Systems of Equations
• How to Graph Single–Variable Inequalities

Step by step guide to solve systems of equations word Problems

• Find the key information in the word problem that can help you define the variables.
• Define two variables: \(x\) and \(y\)
• Write two equations.
• Use the elimination method for solving systems of equations.
• Check the solution by substituting the ordered pair into the original equations.

The Absolute Perfect Practice Workbook for Algebra I

Algebra I for Beginners The Ultimate Step by Step Guide to Acing Algebra I

Systems of equations word problems – example 1:.

Tickets to a movie cost \($8\) for adults and \($5\) for students. A group of friends purchased \(20\) tickets for \($115.00\). How many adult tickets did they buy?

Let \(x\) be the number of adult tickets and \(y\) be the number of student tickets. There are \(20\) tickets. Then: \(x+y=20\). The cost of adults’ tickets is \($8\) and for students ticket is \($5\), and the total cost is \($115\). So, \(8x+5y=115\). Now, we have a system of equations: \(\begin{cases}x+y=20 \\ 8x+5y=115\end{cases}\) Multiply the first equation by \(-5\) and add to the second equation: \(-5(x+y= 20)=- \ 5x-5y=- \ 100\) \(8x+5y+(-5x-5y)=115-100→3x=15→x=5→5+y=20→y=15\). There are \(5\) adult tickets and \(15\) student tickets.

Exercises for Solving Systems of Equations Word Problems

• A farmhouse shelters \(10\) animals, some are pigs and some are ducks. Altogether there are \(36\) legs. How many of each animal are there?
• A class of \(195\) students went on a field trip. They took vehicles, some cars, and some buses. Find the number of cars and the number of buses they took if each car holds \(5\) students and each bus holds \(45\) students.
• The difference of two numbers is \(6\). Their sum is \(14\). Find the numbers.
• The sum of the digits of a certain two–digit number is \(7\). Reversing is increasing the number by \(9\). What is the number?
• The difference of two numbers is \(18\). Their sum is \(66\). Find the numbers.

Download Systems of Equations Word Problems Worksheet

• There are \(8\) pigs and \(2\) ducks.
• There are \(3\) cars and \(4\) buses.
• \(10\) and \(4\).
• \(24\) and \(42\).

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Linear Equations Word Problems Worksheet with Solutions

The first equation family that many students learn about is linear equations. Linear equations have many applications in the real-world, which can make for a really great set of word problems! 

As a student studying algebra, you will encounter many linear equations word problems. That’s why I have put together this linear equations word problems worksheet with solutions!

My hope is that this linear equation word problems worksheet and answer key help you deepen your understanding of linear equations and linear systems!

What Are Linear Equations?

A linear equation is an  algebraic equation  where the highest power on the variable is one. When graphed, a linear equation will produce a straight line.

There are a few ways that we can write linear equations, with two of the most common being slope-intercept form and standard form .

Slope-intercept form is best way to identify the slope of the line and the y-intercept of the line. In general, the equation of a line in slope-intercept form is written as:

In this form,  a  represents the slope of the line and  b  represents the y-intercept of the line.

Equations of lines in standard form are easy to recognize because it is a uniformly recognized form of a line. Standard form allows for easy comparison of coefficients. When two linear equations are in standard form, you can quickly compare the coefficients of x and y.

In general, the standard form of a line is written as:

$$Ax+By=C$$

Note that A and B do not represent the slope of the line or the y-intercept in this form. Instead, A and B are simply constants.

Solving Linear Equations

Any set of word problems relating to linear equations will ask you to solve an equation of some sort. However, there are many different types of solving equations problems that you will encounter as you explore linear equations word problems.

Let’s take a look at a few different types to make sure you know what to expect when you check out the linear equation word problems worksheet with solutions below.

Solving Two-Step Equations and Multi-Step Equations

One of the simplest equation problems that you can solve is a two-step equation. A two-step equation requires you to perform just two steps in order to determine the unique solution to the linear equation.

The first step is to identify the side of the equation with the unknown variable. Your goal will be to isolate this variable (or get it by itself). Consider the following example:

In this example, the variable is on the right-hand side of the equation. To isolate x, we “undo” the operations on the right-hand side of the equation using inverse operations. This just means doing whatever the opposite operation is.

We can apply  order of operations  in reverse to start with the subtraction and then deal with the multiplication. Adding 5 to both sides and then dividing by 2 will result in:

$$\begin{split} 9+5&=2x-5+5 \\ \\ 14 &= 2x \\ \\ \frac{14}{2} &= \frac{2x}{2} \\ \\ 7 &= x \end{split} $$

This shows that the value of the unknown variable here is x = 7. This is a unique solution that will satisfy the equation.

If you want to learn more about finding the solution of linear equations and explore multi-step equations that use the distributive property, check out  these equation solving worksheets !

Systems of Linear Equations

Systems of linear equations are another type of problem that you will see on the linear equations word problems worksheet linked below. A system of linear equations involves two (or more!) linear equations that intersect in some way. 

There are a few different ways that linear equations can intersect :

• Once  at a single point of intersection
• Never  as a result of the lines being parallel 
• Always  as a result of the lines being on top of one another

When solving a system of linear equations, your goal is to determine both unknown variables. If the lines intersect, the solution to the system will be the point of intersection for the lines.

When given a linear equation, we can find the point of intersection between it and a second equation using a few different methods. I made a  video on the substitution method  and a  video on the elimination method  to help you understand these strategies for solving systems before you apply them to word problems involving systems of equations.

What Are Linear Equation Word Problems?

A linear equation word problem involves a real-world situation or scenario that can be solved by setting up and solving linear equations. The equations that are used model the relationships between different quantities in the real-world scenario. 

The topics of these problems vary, ranging from applications in science and physics (ie. calculate the speed of the boat) to business applications (ie. how many sales are required to break even?). The problems that you encounter will also vary in depth and difficulty. 

In my teaching experience, students tend to struggle with word problems because it isn’t always immediately clear what is being asked.  I have seen many students feel very confident in their equation solving skills, yet they still struggle when it comes to solving linear equation word problems.

One reason for this is that you aren’t always given equations from the start while solving word problems.

Tips for Solving Linear Equation Word Problems

During my time as a high school math teacher, I have come across a few tips that I think will help you solve linear equation word problems successfully.

To begin, the first step should always be to define two variables. Read the question carefully and think about the quantities involved. Use variables to represent these quantities.

The second step should be writing an equation that models the scenario. Depending on the problem, you may need to write a second equation as well.

Lastly, think about what the problem is asking you to find. 

For example, are you looking for the values of two unknown variables? If so, you are likely going to be setting up and solving a system of linear equations.

​If you are being asked for the value of a single variable, the chances are you will be solving a single linear equation.

Now that you have a basic understanding of the concepts involved in solving linear equations word problems, it’s time to try a few!

My goal here is to provide you with a worksheet that you can use to practice and feel confident that you understand linear equations word problems!

While I was writing this worksheet, I made sure to include a wide variety of problems that range in difficulty. You will see a few simpler problems involving a two-step equation or multi-step equations, but you will also see a few problems that involve systems of linear equations.

After solving each word problem, be sure to check the answer key to verify that you fully understand the process used to set up the problem and solve it. Reflecting on your understanding is an important part of developing comfort with any given math concept!

Click below to download the linear equations word problems worksheet with solutions!

Using This Linear Equation Word Problems Worksheet

Being able to read a  real-world algebra problem  and set up a linear equation (or a system of linear equations) to solve it is a very challenging skill. In my experience as a math teacher, many students struggle with this concept, even if they fully understand the mathematics that the problem requires.

This is the main reason that I put together this linear equation word problems worksheet with solutions. My goal is to provide you with a set of word problems that you can use to check your understanding of solving linear equations in the real-world.

I hope you found this practice worksheet helpful as you continue your studies of algebra and linear equations!

If you are looking for more linear equations math worksheets in PDF formats, check out my collection of  solving linear inequalities worksheets  and this  linear inequality word problems worksheet .

Did you find this linear equation word problems worksheet with solutions helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!

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Solving Systems of Equations Word Problems Worksheets

We have previously learned about systems of equations. This is when you have multiple linear equations that have the same value for their unknown variables. this allows us to manipulate these equations in many different ways to determine those unknown variables. These systems are evident in many different real-world ways. Students will work with systems of equations that can be found within word problems in this problem set. The worksheets and lessons will show students how to setup up these problems and how to solve them.

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• Carpenter's Project Step-by-step Lesson - We're making chairs, but at what rate and how often?
• Guided Lesson - Making muffins, pearl necklaces, and making sandwiches.
• Guided Lesson Explanation - Create the system first and then find the sweet spot that they both fit into.
• Practice Worksheet - Our scenarios include candle making, buying books and fruit juices.
• Matching Worksheet - Fill in the missing numbers to make each word problem mathematically correct.
• Answer Keys - These are for all the unlocked materials above.

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Some of these problems were so hard to get right that it took several tries.

• Homework 1 - A test has 20 questions worth a total of 100 points. There are two question types on the test. Matching questions are worth 4 points and multiple choice questions are worth 6 points. How many multiple choice and matching questions are there each?
• Homework 2 - 3 small vases and one large vase hold 6 liters of water. If you poured a full large vase into a small vase and filled it, you would still have 2 liters of water left in the large vase. How much water can a large vase hold?
• Homework 3 - Billy went to the fun park. He rode the water slide 5 times and the Ferris wheel twice. He spent $20 on the rides. Jenny rode the water slide 3 times and the Ferris wheel 3 times. Jenny spent $21 on rides. How much does it cost to go on the water slide?

Practice Worksheets

The flow here is set up to make students work a little harder as they go along.

• Practice 1 - Shelly wants to make cookies. She has two different sized cooking pans. She only uses one pan per flavor of cookie. On Saturday, she makes 12 pans of chocolate cookies and 10 pans of vanilla cookies for a total of 312 cookies. On Sunday, She makes 5 pans of chocolate cookies and 3 pans of vanilla cookies for a total of 116 cookies.
• Practice 2 - Ash sells dumplings at the local farmer’s market. On Monday, he sold 12 veggie dumplings and 15 meat dumplings for $69. On Tuesday, he sold 18 veggie dumplings and 7 meat dumplings for $57.
• Practice 3 - The sum of two numbers is 26. The difference of the same two numbers is 8. What are the numbers?

Math Skill Quizzes

The quizzes were hard to get set correctly. It took me a while to get the numbers to balance.

• Quiz 1 - Kelly had a great basketball game today. She made 39 points with 16 different shots. If she only made two and three point shots, how many of each did she make?
• Quiz 2 - At the movies, an adult ticket costs $9 and a kid’s ticket costs $6. If Mr. Fluffy brought 9 people and spent $69, how many kids did he pay for?
• Quiz 3 - A Snickers bar has 150 more calories than a large chocolate chip cookie. Donnie has 3 Snickers bars and 4 chocolate chip cookies for a total of 1,150 calories. How many calories are in each Snickers bar and Chocolate chip cookie?

What Are Systems of Equations in Math?

When you consider systems in mathematics, this normally refers to a series of linear equations that share the same values for their variables. Thus, we must learn the concepts of systems when we are learning algebra. Systems of the equation consist of at least two equations that have the same variables in them. In the concept, the number of equations is directly proportional to the number of variables present in them. Let us take an example which consists of variables x and y. You can communicate these types of problems in algebraic or narrative form.

There are number of different methods you can use to find the values of the variables within these systems. Here is an example of a system of equations: x + y = 6 and -3x + y = 2. Now, we need to solve them simultaneously, to get the values of x and y. When you solve them you will come across a condition. Let us resolve them. x = 6 - y : eq (a)| Here, we will place the value of x in the second equation. -3(6-y) + y = 2 unto -18 + 3y + y = 2 unto 4y = 18 + 2 unto y = 5. Now, we will place the value of y in eq (a). x = 6 - 5 therefore x = 1. The solution set is (1, 5). The condition is that on the graph, both lines will intersect at some point. Your goal by substituting values into the other equation is to locate this point. The best thing about this problem is that it easy to double check your solution. All you need to do is to plug the x and y value of your solution into both equations. If you are correct, they will work for both equations. For the purposes of this topic, we will focus on the narrative form with this topic, but to see it written algebraically is good to get us going.

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• Linear systems – word problems

When it comes to using linear systems to solve word problems, the biggest problem is recognizing the important elements and setting up the equations. Once you do that, these linear systems are solvable just like other linear systems . The same rules apply. The best way to get a grip around these kinds of word problems is through practice, so we will solve a few examples here to get you accustomed to finding elements of linear systems inside of word problems. Example 1 : A farmhouse shelters 16 animals. Some of them are chickens and the others are cows. Altogether these animals have 60 legs. How many chickens and how many cows are in the farmhouse?

First, to make the calculations clearer, we will choose symbols to represent the number of cows and the number of chickens. Let us say that the chickens will be represented with x and the cows with y. Now, this task gave us enough information to make two equations. The first one is that the sum of the number of chickens (x) and the number of cows (y) is 16, since there are only 16 animals in the farmhouse. That equation should look like this: x + y = 16

The second piece of information we have is that the total number of legs in the farmhouse is 60. Since we know that cows have four legs each and chickens have two legs each, we have enough information to make another equation. This one will look like this: 2*x + 4*y = 60

Now we have a system of linear equations with two equations and two variables. The only thing left to do now is to solve the system. We will solve it here for you, but if you need to remind yourself how to do that step by step, read the article called Systems of linear equations . x = 16 – y 32 – 2y + 4y = 60 2y = 28 y = 14 x = 2

We can now see that there are two chickens and 14 cows in the farmhouse. The next example will be a bit harder,

Example 2: Rodney’s Kitchen Supplies makes and sells spoons and forks. It costs the store $2 to buy the supplies needed to make a fork, and $1 for the supplies needed to make a spoon. The store sells the forks for $4 and the spoons for 5$. Last month Rodney’s Kitchen Supplies spent $39 on supplies and sold the all of the forks and spoons that were made last month using those supplies for $93. How many forks and spoons did they make?

As we did in the first example, we will first designate symbols to available variables. So, the number of forks made will be represented with x and the number of spoons with y. Again, we have enough information to make two equations. The total cost of making a particular number of forks (x), which cost $2 to make each, and a particular number of spoons (y), which cost $1 to make each, is $39. So that will be our first equation and it will look like this: 2*x + y = 39 The other piece of information tells us that if we sell that number of forks (x) for $4 each and that number of spoons (y) for $5 each, we will make $93. And that will be our second equation: 4*x + 5*y = 93 This was the hard part. Now all we have to do is to solve this linear system to find how many spoons and how many forks did we make last month. y = 39 – 2x 4x + 5*(39 – 2x) = 93 4x + 195 – 10x = 93 -6x = 93 – 195 -6x = -102 |: (-6) x = 17 y = 5

We can see that last month the store made and sold 17 forks and five spoons.

Although they can seem complicated, mastery and understanding of linear systems and associated word problems will come with a bit of practice. With experience you will be able to recognize their elements and solve even complicated systems with ease. Feel free to use the math worksheets below to practice solving this type of linear systems.

Linear systems – word problems exams for teachers

Linear systems – word problems worksheets for studets

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Systems of Linear Equations Word Problems: Substitution

Use this eighth-grade math worksheet to give learners practice solving word problems by writing and solving systems of equations using substitution! This two-page algebra worksheet will give students valuable practice writing equations to model real-world problems and solving systems of equations using substitution. For more practice solving systems of linear equations word problems, students can also complete the Systems of Linear Equations Word Problems: Graphing and Systems of Linear Equations Word Problems: Elimination worksheets.

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Equation Word Problems Worksheets

This compilation of a meticulously drafted equation word problems worksheets is designed to get students to write and solve a variety of one-step, two-step and multi-step equations that involve integers, fractions, and decimals. These worksheets are best suited for students in grade 6 through high school. Click on the 'Free' icons to sample our handouts.

One Step Equation Word Problem Worksheets

Read and solve this series of word problems that involve one-step equations. Apply basic operations to find the value of unknowns.

(15 Worksheets)

Two-Step Equation Word Problems: Integers

Interpret this set of word problems that require two-step operations to solve the equations. Each printable worksheet has five word problems ideal for 6th grade, 7th grade, and 8th grade students.

• Download the set

Multi-Step Equation Word Problems: Integers

Read each multi-step word problem in these high school pdf worksheets and set up the equation. Solve and find the value of the unknown. More than two steps are required to solve the problems.

Two-Step Equation Word Problems: Fractions and Decimals

Read each word problem and set up the two-step equation. Solve the equation and find the solution. This selection of worksheets includes both fractions and decimals.

Multi-Step Equation Word Problems: Fractions and Decimals

Write multi-step equations that involve both fractions and decimals based on the word problems provided here. Validate your responses with our answer keys.

MCQ - Equation Word Problems

Pick the correct two-step equation that best matches word problems presented here. Evaluate the ability of students to solve two-step equations with this array of MCQ worksheets.

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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.

WORD PROBLEMS ON SYSTEM OF LINEAR EQUATIONS

Grade 8 math worksheets.

Solving systems of linear equations by word problems using both the elimination and substitution methods is a common approach in algebra.

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Before proceeding to word problems, let have a look at equations and know how to solve them.

Steps involved in solving word problems in system of linear equations  

Step 1: Understand the Problem

Carefully read the word problem to understand the context and the relationships between the quantities involved.

Identify the unknowns or variables. Assign clear variables (usually letters) to represent these unknowns.

Make note of any equations or relationships given in the problem.

Step 2: Set Up Equations

Use the information provided in the word problem to set up equations. Translate sentences into mathematical expressions.

Pay attention to keywords like “is,” “equals,” “more than,” “less than,” “twice,” “sum,” “difference,” etc., which indicate mathematical operations.

Create as many equations as necessary to represent the relationships between the variables.

Step 3: Solve the System of Equations  

You now have a system of linear equations derived from the word problem. Depending on the situation, choose one of the following methods to solve the system:

Substitution Method: Solve one equation for one variable and substitute this expression into the other equations.

Elimination Method: Add or subtract equations to eliminate one variable, then solve for the remaining variable.  

Step 4: Solve for the Variables

Solve the system of equations to find the values of the variables. This may involve algebraic manipulation or matrix operations.  

Step 5: Interpret the Solution

Return to the context of the word problem and interpret your solution in real-life terms. What do the values of the variables represent in the given scenario?

Step 6: Check Your Solution

Substitute the values of the variables back into the original equations to ensure they satisfy all the equations. If they do, your solution is correct.

Step 7: Present Your Answer

Write your final answer in a clear and organized manner, often including a sentence or two explaining what the solution means in the context of the problem.

Step 8: Practice

The more word problems you practice, the more skilled you will become in identifying patterns and choosing the appropriate method to solve them. Practice is key to mastering word problems involving simultaneous equations.

Here’s an example to illustrate these steps:

Word Problem:

Susan bought some apples and bananas. She spent a total of $12. Apples cost $2 each, and bananas cost $1 each. If she bought a total of 8 fruits, how many apples and bananas did Susan buy?

Susan is buying apples and bananas.

She spent $12.

Apples cost $2 each, and bananas cost $1 each.

She bought a total of 8 fruits.

Let A represent the number of apples.

Let B represent the number of bananas.

We have two equations:

A + B = 8 (total number of fruits)

2A + 1B = 12 (total cost)

Step 3: Solve the System of Equations

We can use the Elimination Method here:

Multiply the first equation by 2:

2A + 2B = 16

Subtract the second equation from the first:

(2A + 2B) – (2A + 1B) = 16 – 12

Step 4: Solve for the Variables 

We have B = 4. Now, substitute B into the first equation to find A:

Susan bought 4 apples and 4 bananas.

Substitute the values back into the original equations to verify:

4 + 4 = 8 (first equation checks)

2(4) + 1(4) = 12 (second equation checks)

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1) Jim and Kate went to a bookstore. Jim bought 2 novels and 3 magazines for a total of $35, while Kate bought 4 novels and 1 magazine for a total of $45. How much does a novel cost, and how much does a magazine cost?

Let N represent the cost of a novel in dollars, and let M represent the cost of a magazine in dollars.

We have two equations based on the given information:

2N + 3M = 35 (Jim’s purchase)

4N + 1M = 45 (Kate’s purchase)

Now, let’s solve this system of equations. You can use the elimination method or substitution method. Here, we’ll use the substitution method.

From Equation (2), we can express M in terms of N:

M = 45 – 4N

Now, substitute this expression for M into Equation (1):

2N + 3(45 – 4N) = 35  

Simplify and solve for N:

2N + 135 – 12N = 35

Combine like terms:

-10N + 135 = 35

Subtract 135 from both sides:

-10N = -100

Divide by -10:

Now that we know the cost of a novel (N) is $10, we can find the cost of a magazine using Equation (2):

M = 45 – 4(10)

M = 45 – 40

So, a novel costs $10, and a magazine costs $5.

2) Alex and Emily went to a bakery to buy cupcakes and cookies. They bought a total of 12 items and spent $24. Cupcakes cost $2 each, and cookies cost $1 each. How many cupcakes and how many cookies did they buy?

Let C represent the number of cupcakes, and let K represent the number of cookies they bought.

C + K = 12 (total number of items)

2C + 1K = 24 (total cost in dollars)

Now, let’s solve this system of equations using the substitution method:

From Equation (1), we can express C in terms of K:

C = 12 – K

Now, substitute this expression for C into Equation (2):

2(12 – K) + 1K = 24

Simplify and solve for K:

24 – 2K + K = 24

-K + 24 = 24  

Subtract 24 from both sides:

Now, divide by -1:

So, they bought 0 cookies (K = 0). To find the number of cupcakes (C), substitute K = 0 into Equation (1):

Therefore, they bought 12 cupcakes and 0 cookies.

3) Coin Problem:

Sam has a collection of dimes and quarters. He has a total of 12 coins, and the total value of his coins is $2.25. How many dimes and quarters does Sam have?  

Let D represent the number of dimes, and Q represent the number of quarters.

We have the following equations:  

D + Q = 12 (total number of coins)

0.10D + 0.25Q = 2.25 (total value in dollars)

Using the elimination method, multiply the first equation by -0.10:

-0.10D – 0.10Q = -1.20  

Now add the two equations:

(-0.10D – 0.10Q) + (0.10D + 0.25Q) = -1.20 + 2.25

0.15Q = 1.05

Divide by 0.15:

Q = 1.05 / 0.15

Now, substitute Q back into the first equation:

D + 7 = 12  

D = 12 – 7

Sam has 5 dimes and 7 quarters.

4) Alice has some apples and bananas. The total number of fruits she has is 9. If she has 3 more apples than bananas, how many apples and bananas does Alice have?

Let’s use a system of equations to solve this problem.

Let A represent the number of apples, and B represent the number of bananas.

A + B = 9 (total number of fruits is 9)

A = B + 3 (she has 3 more apples than bananas)

Now, we can solve this system of equations. We can use the substitution method:

From Equation (2), we can express A in terms of B:

Now, substitute this expression for A into Equation (1):

(B + 3) + B = 9

Subtract 3 from both sides:

2B = 9 – 3

Divide by 2:

So, Alice has 3 bananas. Now, we can find the number of apples using Equation (2):

Alice has 6 apples and 3 bananas.

5) Investment Problem:

Amy invests $5,000 in two accounts. One account earns an annual interest rate of 4%, and the other earns 5%. After one year, the total interest she earns is $240. How much did she invest in each account?

Let x represent the amount invested at 4%, and y represent the amount invested at 5%.

x + y = 5000 (total amount invested)

0.04x + 0.05y = 240 (total interest earned)

Using the elimination method, multiply the first equation by -0.04:

-0.04x – 0.04y = -200

(-0.04x – 0.04y) + (0.04x + 0.05y) = -200 + 240

0.01y = 40  

Divide by 0.01:

y = 40 / 0.01

Now, substitute y back into the first equation:

x + 4000 = 5000

x = 5000 – 4000

Amy invested $1,000 at 4% and $4,000 at 5%.

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What are word problems involving systems of equations?

Word problems involving systems of equations are real-world scenarios where you need to use multiple equations with multiple variables to find solutions to problems. These problems often require translating verbal information into mathematical equations.

How do I approach solving word problems with systems of equations?

Start by reading the problem carefully to understand the context and relationships between the variables. Then, define variables for the unknowns and create equations based on the problem statement.

What methods can I use to solve word problems with systems of equations?

The common methods include:

• Substitution Method
• Elimination Method

How do I decide which method to use for a specific problem?

The choice of method depends on the problem’s complexity and your preference. Substitution is useful when one equation can be easily solved for one variable. Elimination works when you can eliminate one variable by adding or subtracting equations. Matrix methods are efficient for larger systems, and Cramer’s Rule applies when you have the same number of equations and variables.

Can a word problem with systems of equations have no solution?

Yes, if the equations derived from the problem are inconsistent or contradictory, the system can have no solution.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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Online Systems of Equations Solver

Solve equations and systems of equations with wolfram|alpha, a powerful tool for finding solutions to systems of equations and constraints.

Wolfram|Alpha is capable of solving a wide variety of systems of equations. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Additionally, it can solve systems involving inequalities and more general constraints.

Learn more about:

• Systems of equations

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about solving systems of equations.

• solve y = 2x, y = x + 10
• solve system of equations {y = 2x, y = x + 10, 2x = 5y}
• y = x^2 - 2, y = 2 - x^2
• solve 4x - 3y + z = -10, 2x + y + 3z = 0, -x + 2y - 5z = 17
• solve system {x + 2y - z = 4, 2x + y + z = -2, z + 2y + z = 2}
• solve 4 = x^2 + y^2, 4 = (x - 2)^2 + (y - 2)^2
• x^2 + y^2 = 4, y = x
• View more examples

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Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator

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• Wolfram Problem Generator

What are systems of equations?

A system of equations is a set of one or more equations involving a number of variables..

The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection.

Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent).

More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers.

Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science.