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Linear Equation Word Problems Worksheet

Students will practice solving linear equation word problems.

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linear equations word problems worksheet with answers pdf

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This is a 4 part worksheet:

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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

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

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.

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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.

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MCQ - Two-Step 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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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.

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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.

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Linear Equations Worksheets

Linear equations are equations that have two variables and are a straight line when graphed, based on their slope and y-intercept. Hence, linear equations worksheets have a variety of questions that help students practice key concepts and build a rock-solid foundation of the concepts.

Benefits of Linear Equations Worksheets

Linear equations worksheets are a great resource for students to practice a large variety of problems. These math worksheets are supported by visuals which help students get a crystal clear understanding of the topic. The variety of problems that these worksheets offer helps students approach these concepts in an engaging and fun manner.

Linear equations worksheets come with visual simulation for students to see the problems in action, an answer key that provides a detailed step-by-step solution for students to understand the process better, and a worksheet with detailed solutions.

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Students can practice problems by downloading the linear equations worksheets in PDF format for free.

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Mathematics LibreTexts

1.20: Word Problems for Linear Equations

Word problems are important applications of linear equations. We start with examples of translating an English sentence or phrase into an algebraic expression.

Example 18.1

Translate the phrase into an algebraic expression:

a) Twice a variable is added to 4

Solution: We call the variable \(x .\) Twice the variable is \(2 x .\) Adding \(2 x\) to 4 gives:

\[4 + 2x\nonumber\]

b) Three times a number is subtracted from 7.

Solution: Three times a number is \(3 x .\) We need to subtract \(3 x\) from 7. This means:\

\[7-3 x\nonumber\]

c) 8 less than a number.

Solution: The number is denoted by \(x .8\) less than \(x\) mean, that we need to subtract 8 from it. We get:

\[x-8\nonumber\]

For example, 8 less than 10 is \(10-8=2\).

d) Subtract \(5 p^{2}-7 p+2\) from \(3 p^{2}+4 p\) and simplify.

Solution: We need to calculate \(3 p^{2}+4 p\) minus \(5 p^{2}-7 p+2:\)

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)\nonumber\]

Simplifying this expression gives:

\[\left(3 p^{2}+4 p\right)-\left(5 p^{2}-7 p+2\right)=3 p^{2}+4 p-5 p^{2}+7 p-2 =-2 p^{2}+11 p-2\nonumber\]

e) The amount of money given by \(x\) dimes and \(y\) quarters.

Solution: Each dime is worth 10 cents, so that this gives a total of \(10 x\) cents. Each quarter is worth 25 cents, so that this gives a total of \(25 y\) cents. Adding the two amounts gives a total of

\[10 x+25 y \text{ cents or } .10x + .25y \text{ dollars}\nonumber\]

Now we deal with word problems that directly describe an equation involving one variable, which we can then solve.

Example 18.2

Solve the following word problems:

a) Five times an unknown number is equal to 60. Find the number.

Solution: We translate the problem to algebra:

\[5x = 60\nonumber\]

We solve this for \(x\) :

\[x=\frac{60}{5}=12\nonumber\]

b) If 5 is subtracted from twice an unknown number, the difference is \(13 .\) Find the number.

Solution: Translating the problem into an algebraic equation gives:

\[2x − 5 = 13\nonumber\]

We solve this for \(x\). First, add 5 to both sides.

\[2x = 13 + 5, \text{ so that } 2x = 18\nonumber\]

Dividing by 2 gives \(x=\frac{18}{2}=9\).

c) A number subtracted from 9 is equal to 2 times the number. Find the number.

Solution: We translate the problem to algebra.

\[9 − x = 2x\nonumber\]

We solve this as follows. First, add \(x\) :

\[9 = 2x + x \text{ so that } 9 = 3x\nonumber\]

Then the answer is \(x=\frac{9}{3}=3\)

d) Multiply an unknown number by five is equal to adding twelve to the unknown number. Find the number.

Solution: We have the equation:

\[5x = x + 12.\nonumber\]

Subtracting \(x\) gives

\[4x = 12.\nonumber\]

Dividing both sides by 4 gives the answer: \(x=3\).

e) Adding nine to a number gives the same result as subtracting seven from three times the number. Find the number.

Solution: Adding 9 to a number is written as \(x+9,\) while subtracting 7 from three times the number is written as \(3 x-7\). We therefore get the equation:

\[x + 9 = 3x − 7.\nonumber\]

We solve for \(x\) by adding 7 on both sides of the equation:

\[x + 16 = 3x.\nonumber\]

Then we subtract \(x:\)

\[16 = 2x.\nonumber\]

After dividing by \(2,\) we obtain the answer \(x=8\)

The following word problems consider real world applications. They require to model a given situation in the form of an equation.

Example 18.3

a) Due to inflation, the price of a loaf of bread has increased by \(5 \%\). How much does the loaf of bread cost now, when its price was \(\$ 2.40\) last year?

Solution: We calculate the price increase as \(5 \% \cdot \$ 2.40 .\) We have

\[5 \% \cdot 2.40=0.05 \cdot 2.40=0.1200=0.12\nonumber\]

We must add the price increase to the old price.

\[2.40+0.12=2.52\nonumber\]

The new price is therefore \(\$ 2.52\).

b) To complete a job, three workers get paid at a rate of \(\$ 12\) per hour. If the total pay for the job was \(\$ 180,\) then how many hours did the three workers spend on the job?

Solution: We denote the number of hours by \(x\). Then the total price is calculated as the price per hour \((\$ 12)\) times the number of workers times the number of hours \((3) .\) We obtain the equation

\[12 \cdot 3 \cdot x=180\nonumber\]

Simplifying this yields

\[36 x=180\nonumber\]

Dividing by 36 gives

\[x=\frac{180}{36}=5\nonumber\]

Therefore, the three workers needed 5 hours for the job.

c) A farmer cuts a 300 foot fence into two pieces of different sizes. The longer piece should be four times as long as the shorter piece. How long are the two pieces?

\[x+4 x=300\nonumber\]

Combining the like terms on the left, we get

\[5 x=300\nonumber\]

Dividing by 5, we obtain that

\[x=\frac{300}{5}=60\nonumber\]

Therefore, the shorter piece has a length of 60 feet, while the longer piece has four times this length, that is \(4 \times 60\) feet \(=240\) feet.

d) If 4 blocks weigh 28 ounces, how many blocks weigh 70 ounces?

Solution: We denote the weight of a block by \(x .\) If 4 blocks weigh \(28,\) then a block weighs \(x=\frac{28}{4}=7\)

How many blocks weigh \(70 ?\) Well, we only need to find \(\frac{70}{7}=10 .\) So, the answer is \(10 .\)

Note You can solve this problem by setting up and solving the fractional equation \(\frac{28}{4}=\frac{70}{x}\). Solving such equations is addressed in chapter 24.

e) If a rectangle has a length that is three more than twice the width and the perimeter is 20 in, what are the dimensions of the rectangle?

Solution: We denote the width by \(x\). Then the length is \(2 x+3\). The perimeter is 20 in on one hand and \(2(\)length\()+2(\)width\()\) on the other. So we have

\[20=2 x+2(2 x+3)\nonumber\]

Distributing and collecting like terms give

\[20=6 x+6\nonumber\]

Subtracting 6 from both sides of the equation and then dividing both sides of the resulting equation by 6 gives:

\[20-6=6 x \Longrightarrow 14=6 x \Longrightarrow x=\frac{14}{6} \text { in }=\frac{7}{3} \text { in }=2 \frac{1}{3} \text { in. }\nonumber\]

f) If a circle has circumference 4in, what is its radius?

Solution: We know that \(C=2 \pi r\) where \(C\) is the circumference and \(r\) is the radius. So in this case

\[4=2 \pi r\nonumber\]

Dividing both sides by \(2 \pi\) gives

\[r=\frac{4}{2 \pi}=\frac{2}{\pi} \text { in } \approx 0.63 \mathrm{in}\nonumber\]

g) The perimeter of an equilateral triangle is 60 meters. How long is each side?

Solution: Let \(x\) equal the side of the triangle. Then the perimeter is, on the one hand, \(60,\) and on other hand \(3 x .\) So \(3 x=60\) and dividing both sides of the equation by 3 gives \(x=20\) meters.

h) If a gardener has \(\$ 600\) to spend on a fence which costs \(\$ 10\) per linear foot and the area to be fenced in is rectangular and should be twice as long as it is wide, what are the dimensions of the largest fenced in area?

Solution: The perimeter of a rectangle is \(P=2 L+2 W\). Let \(x\) be the width of the rectangle. Then the length is \(2 x .\) The perimeter is \(P=2(2 x)+2 x=6 x\). The largest perimeter is \(\$ 600 /(\$ 10 / f t)=60\) ft. So \(60=6 x\) and dividing both sides by 6 gives \(x=60 / 6=10\). So the dimensions are 10 feet by 20 feet.

i) A trapezoid has an area of 20.2 square inches with one base measuring 3.2 in and the height of 4 in. Find the length of the other base.

Solution: Let \(b\) be the length of the unknown base. The area of the trapezoid is on the one hand 20.2 square inches. On the other hand it is \(\frac{1}{2}(3.2+b) \cdot 4=\) \(6.4+2 b .\) So

\[20.2=6.4+2 b\nonumber\]

Multiplying both sides by 10 gives

\[202=64+20 b\nonumber\]

Subtracting 64 from both sides gives

\[b=\frac{138}{20}=\frac{69}{10}=6.9 \text { in }\nonumber\]

and dividing by 20 gives

Exit Problem

Write an equation and solve: A car uses 12 gallons of gas to travel 100 miles. How many gallons would be needed to travel 450 miles?

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linear equations word problems worksheet with answers pdf

Graphing linear equations word problems worksheet ( with answer key + pdf)

A linear equation is the equation of a line. The graphs of linear equations are straight lines. Linear equations can be written in several forms. Some word problems require the use of systems of linear equations; there are some clues you need to look for when a word problem requires you to write a system of linear equations.

What is the “Graphing linear equations word problems worksheet (answers+ pdf)”?

The purpose of the “Graphing linear equations word problems worksheet (answers+ pdf)” is to offer students a math worksheet containing linear equation word problems. It will give students a wide variety of opportunities to apply their knowledge in real-world scenarios. This worksheet offers foundational principles of linear equations that will assist students in comprehending the concept.

How will the “Graphing linear equations word problems worksheet (answers+ pdf)”  help you?

This worksheet will help you better understand linear equations and how to plot them on the x-y plane and then utilize them in modeling problems. This worksheet will help students to plot the linear equations by applying each step precisely. If you have a firm understanding of this concept, you can easily handle linear equation graphs.

Instructions on how to use the “Graphing linear equations word problems worksheet (answers+ pdf)”

Technically, you need only two points to create a straight line (linear plot), but plotting three or four points will create a more accurate graph if you’re graphing by hand. In this sheet, you will first learn about the clues you need to look for to apply a system of linear equations and then the steps you need to follow to solve any system of linear equations.

Conclusion 

Linear equations may be written in several forms like slope-Intercept form, horizontal line vertical line, and the general form. Depending on the situation, one can choose any form to plot a system of linear equations. There are several problems in which we can use linear equations to solve them.

If you have any questions or comments, please let us know.

There are several problems which involve relations among known and unknown numbers and can be put in the form of equations. The equations are generally stated in words, and it is for this reason we refer to these problems as word problems. 

Systems of linear equations are often used to evaluate word problems algebraically or graphically. When dealing with complex situations, the graphical technique enables us to tackle equations with non-trivial solutions.

Clues when a word problem requires linear equations:

Steps involved in solving a linear equation word problem:

Linear Equation Graph

The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis. The standard form of a linear equation in one variable is of the form

  ax + b = 0

The graph of a linear equation in two variables x and y forms a straight line. The standard form of a linear equation in two variables is of the form

  ax + by = c. 

There are two ways you can graph linear equations:

Method 1: Using two points to graph a linear equation

The graph of any linear equation like y = x + 3 or y = −3x + 7 is a line, and only two points are needed to define a line. The idea with this method is to find two points on the line by picking values of x.

Graph the linear equation

linear equations word problems worksheet with answers pdf

To find two points on the line, pick any two values of x that would be easy to work with and then find the corresponding value of y

linear equations word problems worksheet with answers pdf

Method 2: Use the slope and y-intercept

A linear equation written in the form y=mx+b is said to be in slope-intercept form. This form shows the slope m and the y-intercept b of the graph. Knowing these two values will let you quickly draw the graph of the linear equation.

linear equations word problems worksheet with answers pdf

Let’s look at the y-intercept first.

The y-intercept is the point where the graph crosses the y-axis (the vertical axis). So, you can plot this point as:

linear equations word problems worksheet with answers pdf

Now consider the slope

linear equations word problems worksheet with answers pdf

Start with the y-intercept and go 3 unit right 

and 2 units up

linear equations word problems worksheet with answers pdf

Connect the points, and you have the graph of your linear function

Step by-step Example

The cost of admission to a popular music concert was $162 for 12 children and 3 adults. The admission was $122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?1 .

Step1 : Understand the problem:

The admission cost for 12 children and 3 adults was $162.

The admission cost for 8 children and 3 adults was $122.

Step 2: Translate the problem to an equation.

Let x represent the admission cost for each child.

Let y represent the admission cost for each adult.

The admission cost for 12 children plus 3 adults is equal to $162 .

That is, 

linear equations word problems worksheet with answers pdf

The admission cost for 8 children plus 3 adults is equal to $122.

linear equations word problems worksheet with answers pdf

Step 3: Carry out the plan and solve the problem.

Subtract the second equation from the first.

Substitute 10 for x in 

linear equations word problems worksheet with answers pdf

Therefore, the cost of admission for each child is $10 and each adult is $14 .

Graphing linear equations word problems worksheet (answers+ pdf)

Problem Solving:

linear equations word problems worksheet with answers pdf

  y = 10x + 600.

Which venue charges less per student? How many students must attend for the total

costs to be the same?

linear equations word problems worksheet with answers pdf

2. Comparing the slopes of the equations, Sunview Resort charges $10 per student, which is less than the $12 per student that Lakeside Inn charges. The total costs are the same for 150 students

Make your own! 

Create one math problem that will make use of the process of linear equations graphical method. Show your solution to the problem you crafted.

linear equations word problems worksheet with answers pdf

Reflection:

What are your thoughts on the use of linear equations in word problems? Was there any struggle or difficulty you experienced while converting them?

linear equations word problems worksheet with answers pdf

You can download this worksheet here.

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Honors algebra 2 worksheet word problems linear equations answers

View Word Problems Linear and Inequalities.pdf from MATH 101 at Mallard Creek High. Honors Algebra II Worksheet: Word Problems: Linear Equations Assign

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    Linear Equation Word Problems Worksheet (pdf) and Answer Key. 31 scaffolded questions on simplifying imaginary numbers Home Worksheets Linear Equations Word Problems Linear Equation Word Problems Worksheet Mathworksheetsgo.com is now a part of Mathwarehouse.com. All of your worksheets are now here on Mathwarehouse.com. Please update your bookmarks!

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    Water flows into the tank at a rate of 8 gallons per minute. Write a linear equation to model this situation. Find the volume of water in the tank 25 minutes after Baxter begins filling the tank. 10. A video rental store charges a $20 membership fee and $2.50 for each video rented. Write and graph a linear equation (y=mx+b) to model this situation.

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    Download Linear Equations Word Problems Worksheet PDFs These math worksheets should be practiced regularly and are free to download in PDF formats. ☛ Check Grade wise Linear Equations Word Problems Worksheets 8th Grade Linear Equations Worksheets Explore math program

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    SECTION 3.1: LINEAR EQUATIONS A. VERIFYING SOLUTIONS A linear equation is made up of two expressions that are equal to each other. A linear equation may have one or two variables in it, where each variable is raised to the power of 1. No variable in a linear equation can have a power greater than 1. Linear equation: 2𝑦𝑦= 3𝑥𝑥+ 1

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  17. 1.20: Word Problems for Linear Equations

    We solve this as follows. First, add x : 9 = 2 x + x so that 9 = 3 x Then the answer is x = 9 3 = 3 d) Multiply an unknown number by five is equal to adding twelve to the unknown number. Find the number. Solution: We have the equation: 5 x = x + 12. Subtracting x gives 4 x = 12. Dividing both sides by 4 gives the answer: x = 3.

  18. PDF Systems of Equations Word Problems

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  19. Graphing linear equations word problems worksheet ( with answer key + pdf)

    The equations are generally stated in words, and it is for this reason we refer to these problems as word problems. Systems of linear equations are often used to evaluate word problems algebraically or graphically. When dealing with complex situations, the graphical technique enables us to tackle equations with non-trivial solutions. Clues when ...

  20. Honors algebra 2 worksheet word problems linear equations answers

    Honors Algebra II Summer Assignment 2018. Honors Algebra 2. Periods 1, 9 Thursday 5/2 - Practice Day KEY BLANK WORKSHEET Thursday 11/15 - Go over word problems/ Review #1 HERE ANSWER KEY.

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    4.5 Linear Models - Word Problems. Write an equation, in slope-intercept form, to model each situation. 1. You rent a bicycle for $20 plus $2 per hour. y=2x+20.

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  24. Mr Barton Maths

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