solving systems of equations by elimination word problems

Math Worksheets

Just another wordpress site, site navigation.

• Constant of Proportionality Worksheets
• Coordinate Graph Worksheets–Interpreting
• Equivalent Expressions Worksheets–Perimeter
• Equivalent Expressions Worksheets–Word Problems
• Integer Division Worksheets
• Number Line Worksheets–Integers
• Number Line Worksheets–Rational Numbers
• Rational Number Expressions Worksheets
• Tape Diagram Worksheets
• Analyzing Equations Worksheets
• Function Interval Worksheets
• Repeating Decimals Worksheets
• Scientific Notation Worksheets–Multiples
• Simultaneous Linear Equation Worksheets (Part I)
• Simultaneous Linear Equation Worksheets (Part II)
• Systems of Equations (How Many Solutions?)
• Transformation Effects Worksheets
• Transformation Series Worksheets
• Evaluating Expressions Worksheets
• Factoring Polynomials Worksheets
• Graphing Inequalities Worksheets (Single Variable)
• Solving Inequalities Worksheets
• Solving Inequalities with Absolute Value Worksheets
• Order of Operations Worksheets
• Equations & Word Problems (Combining Like Terms)
• Slope of a Line from a Graph–Points Given
• Slope Worksheets (Two Points-No Graph)
• Changing One Equation
• Changing Two Equations

Word Problems

• Multiple Choice Worksheets
• Substitution Worksheets
• Already Graphed
• Graphing Systems of Equations (Slope-Intercept Form)
• Graphing Systems of Equations (Standard Form)
• Trigonometry Worksheets
• Auto-Generated Worksheets
• Interpreting Points on a Graph
• Adding Decimals Worksheets
• Comparing Decimals-Decimal Inequalities
• Decimal Division-Integer Divisors (1 Digit)
• Decimal Division-Integer Divisors (2 Digit)
• Decimal Division-Integer Divisors (3 Digit)
• Decimal Division-Integer Divisors (MIXED Digits)
• Decimal Division-Decimal Divisors (Tenths into Tenths)
• Decimal Division-Decimal Divisors (Tenths into Hundredths)
• Decimal Division-Decimal Divisors (Tenths into Thousandths)
• Decimal Division-Decimal Divisors (Hundredths into Hundredths)
• Decimal Division-Decimal Divisors (Hundredths into Thousandths)
• Decimal Division-Decimal Divisors (Thousandths into Thousandths)
• Decimal Division-MIXED Decimal Divisors
• MIXED Decimal & Integer Divisors (1-Digit)
• MIXED Decimal & Integer Divisors (2-Digits)
• MIXED Decimal & Integer Divisors (3-Digits)
• Adding 1 Zero (Single-Digit Integer Divisor)
• Adding 1 Zero (Two-Digit Integer Divisor)
• Adding 1 Zero (Single-Digit Decimal Divisors)
• Adding 1 Zero (Two-Digit Decimal Divisors)
• Adding 2 Zeros (Single-Digit Integer Divisors)
• Adding 2 Zeros (Two-Digit Integer Divisors)
• Adding 2 Zeros (Decimal Divisors)
• Repeating Decimals (One Digit Patterns)
• Repeating Decimals (Two Digit Patterns)
• Repeating Decimals (Three Digit Patterns)
• Decimal Division Word Problem Worksheets
• Multiplying Decimals Worksheets
• Subtracting Decimals Worksheets
• Writing Decimals as Fractions Worksheets
• Checking Equation Solutions–Distributive Property
• Checking Equation Solutions–Like Terms
• Checking Equation Solutions–Variables on Both Sides
• Checking Two-Step Equation Solutions
• Solving Equations with Like Terms
• Solving Equations with the Distributive Property Worksheets
• Solving Equations with Variables on Both Sides Worksheets
• Solving Equations with Absolute Value Worksheets
• Solving Proportions
• Equations and Word Problems (Two Step Equations)
• Equations and Word Problems (Combining Like Terms) Worksheets
• Adding Fractions Worksheets
• Comparing Fractions Worksheets
• Dividing Fractions Worksheets
• Multiplying Fractions Worksheets
• Proportions & Fractions Worksheets
• Subtracting Fractions Worksheets
• Exterior Angles Worksheets
• Interior Angles Worksheets
• Parallel Lines & Transversals Worksheets
• Areas of Circles Worksheets
• Areas of Parallelograms Worksheets
• Areas of Trapezoids Worksheets
• Areas of Triangles Worksheets
• Radius Given (Using 3.14)
• Diameter Given (Using 3.14)
• Radius or Diameter Given (Using 3.14)
• Radius Given (In Terms of Pi)
• Diameter Given (In Terms of Pi)
• Radius or Diameter Given (In Terms of Pi)
• Volume of a Rectangular Prism
• Volume of a Triangular Prism
• Absolute Value of a Number Worksheets
• Absolute Value Expressions (Simplifying) Worksheets
• Absolute Value Equations Workssheets
• Absolute Value Inequalities Worksheets
• Addition Worksheets
• Division Worksheets
• Multiplication Worksheets
• Percentage Worksheets
• Square Roots
• Subtraction Worksheets
• Mean/Median/Mode/Range Worksheets
• Mean Worksheets
• Median Worksheets
• Graphs and Mean/Median/Mode Worksheets
• Absolute Value–Simplifying Expressions Worksheets
• Absolute Value Equations Worksheets
• Absolute Value Inequality Worksheets
• Probability & Compound Events Worksheets
• Probability & Predictions Worksheets
• Theoretical Probability Worksheets
• Comparing Ratios Word Problem Worksheets
• Comparing Ratios Worksheets
• Part-to-Part Ratio Worksheets
• Part-to-Whole Ratio Worksheets
• Ratio Word Problems (w/Fractions)
• Simplified Ratios Word Problem Worksheets
• Writing Ratios Word Problem Worksheets
• Writing Ratios Word Problems (missing info)
• Writing Ratios Word Problems (w/distractors)
• Writing Ratios Worksheets
• Comparing Unit Rates Worksheets
• Unit Rate Word Problem Worksheets
• Unit Rates & Graphs Worksheets
• Unit Rates & Proportions Worksheets
• Unit Rates & Tables

Looking for Something?

Popular content.

• Simplifying Expressions Worksheets
• Absolute Value Worksheets

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?

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

• + ACCUPLACER Mathematics
• + ACT Mathematics
• + AFOQT Mathematics
• + ALEKS Tests
• + ASVAB Mathematics
• + ATI TEAS Math Tests
• + Common Core Math
• + DAT Math Tests
• + FSA Tests
• + FTCE Math
• + GED Mathematics
• + Georgia Milestones Assessment
• + GRE Quantitative Reasoning
• + HiSET Math Exam
• + HSPT Math
• + ISEE Mathematics
• + PARCC Tests
• + Praxis Math
• + PSAT Math Tests
• + PSSA Tests
• + SAT Math Tests
• + SBAC Tests
• + SIFT Math
• + SSAT Math Tests
• + STAAR Tests
• + TABE Tests
• + TASC Math
• + TSI Mathematics
• + ACT Math Worksheets
• + Accuplacer Math Worksheets
• + AFOQT Math Worksheets
• + ALEKS Math Worksheets
• + ASVAB Math Worksheets
• + ATI TEAS 6 Math Worksheets
• + FTCE General Math Worksheets
• + GED Math Worksheets
• + 3rd Grade Mathematics Worksheets
• + 4th Grade Mathematics Worksheets
• + 5th Grade Mathematics Worksheets
• + 6th Grade Math Worksheets
• + 7th Grade Mathematics Worksheets
• + 8th Grade Mathematics Worksheets
• + 9th Grade Math Worksheets
• + HiSET Math Worksheets
• + HSPT Math Worksheets
• + ISEE Middle-Level Math Worksheets
• + PERT Math Worksheets
• + Praxis Math Worksheets
• + PSAT Math Worksheets
• + SAT Math Worksheets
• + SIFT Math Worksheets
• + SSAT Middle Level Math Worksheets
• + 7th Grade STAAR Math Worksheets
• + 8th Grade STAAR Math Worksheets
• + THEA Math Worksheets
• + TABE Math Worksheets
• + TASC Math Worksheets
• + TSI Math Worksheets
• + AFOQT Math Course
• + ALEKS Math Course
• + ASVAB Math Course
• + ATI TEAS 6 Math Course
• + CHSPE Math Course
• + FTCE General Knowledge Course
• + GED Math Course
• + HiSET Math Course
• + HSPT Math Course
• + ISEE Upper Level Math Course
• + SHSAT Math Course
• + SSAT Upper-Level Math Course
• + PERT Math Course
• + Praxis Core Math Course
• + SIFT Math Course
• + 8th Grade STAAR Math Course
• + TABE Math Course
• + TASC Math Course
• + TSI Math Course
• + Number Properties Puzzles
• + Algebra Puzzles
• + Geometry Puzzles
• + Intelligent Math Puzzles
• + Ratio, Proportion & Percentages Puzzles
• + Other Math Puzzles

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 \(42\).

The Best Books to Ace Algebra

Algebra I Practice Workbook The Most Comprehensive Review of Algebra 1

Algebra ii for beginners the ultimate step by step guide to acing algebra ii, pre-algebra for beginners the ultimate step by step guide to preparing for the pre-algebra test, pre-algebra tutor everything you need to help achieve an excellent score.

by: Effortless Math Team about 4 years ago (category: Articles , Free Math Worksheets )

Effortless Math Team

Related to this article, more math articles.

• How Can Redefining a Function’s Value Solve Your Limit Problems
• Best Laptops for Math Majors in 2024
• The Ultimate 7th Grade FSA Math Course (+FREE Worksheets)
• The Ultimate TNReady Algebra 1 Course (+FREE Worksheets)
• Top 10 4th Grade MAP Math Practice Questions
• Best Desktop Computers For Online Math Teachers
• Top 10 Tips to ACE the SAT Mathematics
• 5th Grade STAAR Math FREE Sample Practice Questions
• HSPT Math-Test Day Tips
• 5th Grade MCAS Math Worksheets: FREE & Printable

What people say about "How to Solve Systems of Equations Word Problems? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.

Leave a Reply Cancel reply

You must be logged in to post a comment.

The Ultimate Algebra Bundle From Pre-Algebra to Algebra II

• ATI TEAS 6 Math
• ISEE Upper Level Math
• SSAT Upper-Level Math
• Praxis Core Math
• 8th Grade STAAR Math

Limited time only!

Save Over 45 %

It was $89.99 now it is $49.99

Login and use all of our services.

Effortless Math services are waiting for you. login faster!

Register Fast!

Password will be generated automatically and sent to your email.

After registration you can change your password if you want.

• Math Worksheets
• Math Courses
• Math Topics
• Math Puzzles
• Math eBooks
• GED Math Books
• HiSET Math Books
• ACT Math Books
• ISEE Math Books
• ACCUPLACER Books
• Premium Membership
• Youtube Videos

Effortless Math provides unofficial test prep products for a variety of tests and exams. All trademarks are property of their respective trademark owners.

• Bulk Orders
• Refund Policy

• Pre-Algebra Topics
• Algebra Topics
• Algebra Calculator
• Algebra Cheat Sheet
• Algebra Practice Test
• Algebra Readiness Test
• Algebra Formulas
• Want to Build Your Own Website?

Sign In / Register

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

• System of Equations
• Systems Word Problems

Need More Help With Your Algebra Studies?

Get access to hundreds of video examples and practice problems with your subscription! 

Click here for more information on our affordable subscription options.

Not ready to subscribe?  Register for our FREE Pre-Algebra Refresher course.

ALGEBRA CLASS E-COURSE MEMBERS

Click here for more information on our Algebra Class e-courses.

Need Help? Try This Online Calculator!

Affiliate Products...

On this site, I recommend only one product that I use and love and that is Mathway   If you make a purchase on this site, I may receive a small commission at no cost to you. 

Privacy Policy

Let Us Know How we are doing!

  send us a message to give us more detail!

Would you prefer to share this page with others by linking to it?

• Click on the HTML link code below.
• Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.

Copyright © 2009-2020   |   Karin Hutchinson   |   ALL RIGHTS RESERVED.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

5.2: Solve Systems of Equations by Substitution

• Last updated
• Save as PDF
• Page ID 15152

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Learning Objectives

By the end of this section, you will be able to:

• Solve a system of equations by substitution
• Solve applications of systems of equations by substitution

Before you get started, take this readiness quiz.

• Simplify −5(3−x). If you missed this problem, review Example 1.10.43 .
• Simplify 4−2(n+5). If you missed this problem, review Example 1.10.49 .
• Solve for y. 8y−8=32−2y If you missed this problem, review Example 2.3.22 .
• Solve for x. 3x−9y=−3 If you missed this problem, review Example 2.6.22 .

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

\(\left\{\begin{array}{l}{2 x+y=7} \\ {x-2 y=6}\end{array}\right.\)

We will first solve one of the equations for either x or y . We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in Example \(\PageIndex{1}\).

Example \(\PageIndex{1}\): How to Solve a System of Equations by Substitution

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x+y=7} \\ {x-2 y=6}\end{array}\right.\)

Try It \(\PageIndex{2}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{-2 x+y=-11} \\ {x+3 y=9}\end{array}\right.\)

Try It \(\PageIndex{3}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+3 y=10} \\ {4 x+y=18}\end{array}\right.\)

SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION.

• Solve one of the equations for either variable.
• Substitute the expression from Step 1 into the other equation.
• Solve the resulting equation.
• Substitute the solution in Step 3 into one of the original equations to find the other variable.
• Write the solution as an ordered pair.
• Check that the ordered pair is a solution to both original equations.

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example \(\PageIndex{4}\).

Example \(\PageIndex{4}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+y=-1} \\ {y=x+5}\end{array}\right.\)

The second equation is already solved for y . We will substitute the expression in place of y in the first equation.

Try It \(\PageIndex{5}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+y=6} \\ {y=3 x-2}\end{array}\right.\)

Try It \(\PageIndex{6}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x-y=1} \\ {y=-3 x-6}\end{array}\right.\)

(−1,−3)

If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y .

Example \(\PageIndex{7}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{3 x+y=5} \\ {2 x+4 y=-10}\end{array}\right.\)

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

Try It \(\PageIndex{8}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x+y=2} \\ {3 x+2 y=-1}\end{array}\right.\)

(1,−2)

Try It \(\PageIndex{9}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{-x+y=4} \\ {4 x-y=2}\end{array}\right.\)

In Example \(\PageIndex{7}\) it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example \(\PageIndex{10}\) it will be easier to solve for x .

Example \(\PageIndex{10}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-2 y=-2} \\ {3 x+2 y=34}\end{array}\right.\)

We will solve the first equation for \(x\) and then substitute the expression into the second equation.

Try It \(\PageIndex{11}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-5 y=13} \\ {4 x-3 y=1}\end{array}\right.\)

(−2,−3)

Try It \(\PageIndex{12}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-6 y=-6} \\ {2 x-4 y=4}\end{array}\right.\)

When both equations are already solved for the same variable, it is easy to substitute!

Example \(\PageIndex{13}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=-2 x+5} \\ {y=\frac{1}{2} x}\end{array}\right.\)

Since both equations are solved for y , we can substitute one into the other.

Try It \(\PageIndex{14}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=3 x-16} \\ {y=\frac{1}{3} x}\end{array}\right.\)

Try It \(\PageIndex{15}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=-x+10} \\ {y=\frac{1}{4} x}\end{array}\right.\)

Be very careful with the signs in the next example.

Example \(\PageIndex{16}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x+2 y=4} \\ {6 x-y=8}\end{array}\right.\)

We need to solve one equation for one variable. We will solve the first equation for y .

Try It \(\PageIndex{17}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-4 y=-4} \\ {-3 x+4 y=0}\end{array}\right.\)

\((2,\frac{3}{2})\)

Try It \(\PageIndex{18}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x-y=0} \\ {2 x-3 y=5}\end{array}\right.\)

\((−\frac{1}{2},−2)\)

In Example , it will take a little more work to solve one equation for x or y .

Example \(\PageIndex{19}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x-3 y=6} \\ {15 y-20 x=-30}\end{array}\right.\)

We need to solve one equation for one variable. We will solve the first equation for x .

Try It \(\PageIndex{20}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x-3 y=12} \\ {-12 y+8 x=48}\end{array}\right.\)

infinitely many solutions

Try It \(\PageIndex{21}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x+2 y=12} \\ {-4 y-10 x=-24}\end{array}\right.\)

Look back at the equations in Example \(\PageIndex{22}\). Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Example \(\PageIndex{22}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x-2 y=-10} \\ {y=\frac{5}{2} x}\end{array}\right.\)

The second equation is already solved for y , so we can substitute for y in the first equation.

Try It \(\PageIndex{23}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{3 x+2 y=9} \\ {y=-\frac{3}{2} x+1}\end{array}\right.\)

no solution

Try It \(\PageIndex{24}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x-3 y=2} \\ {y=\frac{5}{3} x-4}\end{array}\right.\)

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

HOW TO USE A PROBLEM SOLVING STRATEGY FOR SYSTEMS OF LINEAR EQUATIONS.

• Read the problem. Make sure all the words and ideas are understood.
• Identify what we are looking for.
• Name what we are looking for. Choose variables to represent those quantities.
• Translate into a system of equations.
• Solve the system of equations using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example \(\PageIndex{25}\)

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Try It \(\PageIndex{26}\)

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

The numbers are 3 and 7.

Try It \(\PageIndex{27}\)

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

The numbers are 2 and −8.

In the Example \(\PageIndex{28}\), we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

Example \(\PageIndex{28}\)

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and width of the rectangle.

Try It \(\PageIndex{29}\)

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

The length is 12 and the width is 8.

Try It \(\PageIndex{30}\)

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

The length is 23 and the width is 6.

For Example \(\PageIndex{31}\) we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

Example \(\PageIndex{31}\)

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

We will draw and label a figure.

Try It \(\PageIndex{32}\)

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of the angles are 22 degrees and 68 degrees.

Try It \(\PageIndex{33}\)

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

The measure of the angles are 36 degrees and 54 degrees.

Example \(\PageIndex{34}\)

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

Try It \(\PageIndex{35}\)

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

There would need to be 160 policies sold to make the total pay the same.

Try It \(\PageIndex{36}\)

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

Kenneth would need to sell 1,000 suits.

Access these online resources for additional instruction and practice with solving systems of equations by substitution.

• Instructional Video-Solve Linear Systems by Substitution
• Instructional Video-Solve by Substitution

Key Concepts

Elimination Calculator

gives you step-by-step help on solving systems by elimination.

What do you want to calculate?

• Solve for Variable
• Practice Mode
• Step-By-Step

Example (Click to try)

• Enter your equations separated by a comma in the box, and press Calculate!
• Or click the example.

About Elimination

Need more problem types? Try MathPapa Algebra Calculator

Clear Elimination Calculator »

WORD PROBLEMS USING ELIMINATION METHOD

Problem 1 :

Find the value of two numbers if their sum is 12 and their difference is 4.

Let x and y be the two numbers.

Given : T heir sum is 12.

x + y = 12 ----(1)

Given : T heir difference is 4.

x - y = 4 ----(1)

(1) + (2) :

Divide both sides by 2.

Substitute x = 8 into (1).

Subtract 12 from both sides.

Therefore, the numbers are 8 and 4.

Problem 2 :

Kevin bought pen and pencils in a total of 50. The cost of each pen is $2 and that of a pencil is $1.50. If he had paid a total of $85 for his purchase, find the cost of a pen and a pencil.

Let x and y be the costs of a pen and a pencil respectively.

Given : Kevin bought pen and pencils in a total of 50.

x + y = 50 ----(1)

Given : The cost of each pen is $2, a pencil is  $1.50 and Kevin paid a total of $85 for his purchase.

2x + 1.5y = 85 ----(2)

1.5(1) - (2) :

Divide both sides by 0.5.

Substitute x = 20 into (1).

20 + y = 50

Subtract 20 from both sides.

number of pens = 20

number of pencils = 30

Problem 3 :

The sum of two numbers is 7. The difference between 5 times the larger and 3 times the smaller is equal 11. Find the numbers. 

Let x and y be the two numbers such that x > y.

Given : The sum of two numbers is 7.

x + y = 7 ----(1)

Given : The difference between 5 times the larger and 3 times the smaller is equal 11.

5x - 3y = 11 ----(2)

3(1) + (2) :

Divide both sides by 8.

Substitute x = 4 into (1).

Subtract 3 from both sides.

Therefore, the two numbers are 4 and 3.

Problem 4 :

Chase and Sara went to the candy store. Chase bought 5 pieces of fudge and 3 pieces of bubble gum for a total of $5.70. Sara bought 2 pieces of fudge and 10 pieces of bubble gum for a total of $3.60. Find the cost of 1 piece of fudge and 1 piece of bubble gum?

Let f and g be the costs of 1 piece of fudge and 1 piece of bubble.

From the given information,

5f + 3b = 5.7 ----(1)

2f + 10b = 3.6 ----(2)

2(1) - 5(2) :

Divide both sides by -44.

Substitute x = 0.15 into (2).

 2(0.15) + 10b = 3.6

0.3 + 10b = 3.6

Subtract 0.3 from both sides.

Divide both sides by 10.

cost of 1 piece of fudge = $0.15

cost of 1  piece of bubble gum = $0.33

Problem 5 :

Daily earnings of Oliver and Henry are in the ratio 3 : 4 and their expenditures are in the ratio 5 : 7. If each saves $50 per day, find the daily earnings of each.

From the earnings ratio 3 : 4,

earnings of Oliver = 3x

earnings of Henry = 4x

From the expenditures ratio 5 : 7,

expenditure of Oliver = 5y

expenditure of Henry = 7y

The relationship between income, expenditure and savings :

Income - Expenditure = Savings

Then, we have

3x - 5y = 50 ----(1)

4x - 7y = 50 ----(2)

7(1) - 5(2) :

earnings of Oliver = 3(100) = $300

earnings of Henry = 4(100) = $400

Problem 6 :

The sum of the present ages of a father and his son is 45 years. 10 years hence, the difference between their ages is 25 years. Find the present age of the father and son.

Let f and s be the present ages of father and son.

Given : The sum of the present ages 45.

f + s = 45 ----(1)

Given : 10 years hence, the difference between the ages is 25.

(f + 10) - (s + 10) = 25

f + 10 - s - 10 = 25

f - s = 25 ----(2)

Substitute f = 35 into (1).

 35 + s = 45

Subtract 35 from both sides.

present age of the father = 35 years

present age of the son = 10 years

Problem 7 :

In a three digit number, the middle digit is zero and sum of the other two digits is 9. If 99 is added to it, the digits are reversed. Find the three digit number.

Let x 0 y  be the required three digit number.

x + y = 9 ----(1)

Given : When 99  is added to it, the digits are reversed.

x 0 y + 99 = y 0 x

100(x) + 10(0) + 1(y) + 99 = 100(y) + 10(0) + 1(x)

100x + 0 + y + 99 = 100y + 0 + x

100x + y + 99 = 100y + x

99x - 99y = -99

Divide both sides by 99.

x - y = -1 ----(2)

Substitute x  = 4 into (1).

Subtract 4 from both sides.

x 0 y = 405

Therefore, the three digit number is 405.

Problem 8 :

Two numbers are such that twice the greater number exceeds twice the sm aller one by 18. One-third of the smaller number and one-fifth of the greater number are together 21. Find the two numbers.

Let x  and  y  be the two numbers such that x > y .

Given : Twice the greater number exceeds twice the smaller one by 18.

2x - 2y = 18

x - y = 9 ----(1)

Given : One-third of the smaller number and one-fifth of the greater number are together 21.

5y  + 3x = 315

3x + 5y = 315 ----(2)

5(1) + (2) :

Substitute x  = 45 into (1).

Subtract 45 from both sides.

Therefore, the two numbers are 45 and 36.

Kindly mail your feedback to   [email protected]

We always appreciate your feedback.

© All rights reserved. onlinemath4all.com

• Sat Math Practice
• SAT Math Worksheets
• PEMDAS Rule
• BODMAS rule
• GEMDAS Order of Operations
• Math Calculators
• Transformations of Functions
• Order of rotational symmetry
• Lines of symmetry
• Compound Angles
• Quantitative Aptitude Tricks
• Trigonometric ratio table
• Word Problems
• Times Table Shortcuts
• 10th CBSE solution
• PSAT Math Preparation
• Privacy Policy
• Laws of Exponents

Recent Articles

How to Convert Between Polar and Rectangular Coordinates

May 31, 24 08:11 PM

How to Convert Between Polar and Rectangular Equations

May 31, 24 08:05 PM

Number Line

• elimination\:x+y+z=25,\:5x+3y+2z=0,\:y-z=6
• elimination\:x+2y=2x-5,\:x-y=3
• elimination\:5x+3y=7,\:3x-5y=-23
• elimination\:x+z=1,\:x+2z=4

elimination-system-of-equations-calculator

• High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,...

Please add a message.

Message received. Thanks for the feedback.