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Linear Functions Problems with Solutions

Linear functions are highly used throughout mathematics and are therefore important to understand. A set of problems involving linear functions , along with detailed solutions, are presented. The problems are designed with emphasis on the meaning of the slope and the y intercept.

Problem 1: f is a linear function. Values of x and f(x) are given in the table below; complete the table.

Problem 2: A family of linear functions is given by

Solution to Problem 2: a)

Problem 3: A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P ; grows as a linear function of time t, where t is the number of years after 2003. a) How many students will be enrolled in the school in 2010? b) Find a linear function that relates the student population to the time t. Solution to Problem 3: a) The given information may be written as ordered pairs (t , P). The year 2003 correspond to t = 0 and the year 2006 corresponds to t = 3, hence the 2 ordered pairs (0, 1200) and (3, 1500) Since the population grows linearly with the time t, we use the two ordered pairs to find the slope m of the graph of P as follows m = (1500 - 1200) / (6 - 3) = 100 students / year The slope m = 100 means that the students population grows by 100 students every year. From 2003 to 2010 there are 7 years and the students population in 2010 will be P(2010) = P(2003) + 7 * 100 = 1200 + 700 = 1900 students. b) We know the slope and two points, we may use the point slope form to find an equation for the population P as a function of t as follows P - P1 = m (t - t1) P - 1200 = 100 (t - 0) P = 100 t + 1200

Problem 4: The graph shown below is that of the linear function that relates the value V (in $) of a car to its age t, where t is the number of years after 2000.

Problem 5: The cost of producing x tools by a company is given by

Problem 6: A 500-liter tank full of oil is being drained at the constant rate of 20 liters par minute. a) Write a linear function V for the number of liters in the tank after t minutes (assuming that the drainage started at t = 0). b) Find the V and the t intercepts and interpret them. e) How many liters are in the tank after 11 minutes and 45 seconds? Solution to Problem 6: After each minute the amount of oil in the tank deceases by 20 liters. After t minutes, the amount of oil in the tank decreases by 20*t liters. Hence if at the start there 500 liters, after t minute the amount V of oil left in the tank is given by V = 500 - 20 t b) To find the V intercept, set t = 0 in the equation V = 500 - 20 t. V = 500 liters : it is the amount of oil at the start of the drainage. To find the t intercept, set V = 0 in the equation V = 500 - 20 t and solve for t. 0 = 500 - 20 t t = 500 / 20 = 25 minutes : it is the total time it takes to drain the 500 liters of oil. c) Convert 11 minutes 45 seconds in decimal form. t = 11 minutes 45 seconds = 11.75 minutes Calculate V at t = 11.75 minutes. V(11.75) = 500 - 20*11.75 = 265 liters are in the tank after 11 minutes 45 seconds of drainage.

Problem 7: A 50-meter by 70-meter rectangular garden is surrounded by a walkway of constant width x meters.

Problem 8: A driver starts a journey with 25 gallons in the tank of his car. The car burns 5 gallons for every 100 miles. Assuming that the amount of gasoline in the tank decreases linearly, a) write a linear function that relates the number of gallons G left in the tank after a journey of x miles. b) What is the value and meaning of the slope of the graph of G? c) What is the value and meaning of the x intercept? Solution to Problem 8: a) If 5 gallons are burnt for 100 miles then (5 / 100) gallons are burnt for 1 mile. Hence for x miles, x * (5 / 100) gallons are burnt. G is then equal to the initial amount of gasoline decreased by the amount gasoline burnt by the car. Hence G = 25 - (5 / 100) x b) The slope of G is equal to 5 / 1000 and it represent the amount of gasoline burnt for a distance of 1 mile. c) To find the x intercept, we set G = 0 and solve for x. 25 - (5 / 100) x = 0 x = 500 miles : it is the distance x for which all 25 gallons of gasoline will be burnt.

Problem 9: A rectangular wire frame has one of its dimensions moving at the rate of 0.5 cm / second. Its width is constant and equal to 4 cm. If at t = 0 the length of the rectangle is 10 cm,

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Linear Functions Worksheets

This collection of linear functions worksheets is a complete package and leaves no stone unturned. Eighth grade and high school students gain practice in identifying and distinguishing between a linear and a nonlinear function presented as equations, graphs and tables. Identify the function rule, complete tables, evaluate, graph, compare and transform linear functions are some topics dealt with here. Start off your functions practice with our free worksheets!

Identify Linear and Nonlinear Functions from Equation

Try this set of linear vs nonlinear functions worksheet pdfs to determine whether a function is linear or not. If the equation can be written in the slope-intercept form, y=mx+b then it is linear.

• Download the set

Is the Function Linear or Nonlinear? | Graph

8th grade students learn to distinguish between linear and nonlinear functions by observing the graphs. The graph of a linear function is a straight line, while the graph of a nonlinear function is a curve.

Is the Function Linear or Nonlinear? | Table

Examine the input(x) and output(y) values of the table inthese linear function worksheets for grade 8. If the rate of change for y with respect to x remains constant, then the table represents a linear function.

Linear Function | Level 1

Substitute the x-values in the linear expression to find the y-values in each function table worksheet. The easy level deals with integers, while the moderate level focuses on fractions and decimals.

Linear Function | Level 2

Completing the function table in this batch of pdf worksheets comes with a twist. Students are expected to plug in either the input or output values in the function rule to determine the missing value.

Finding the Function Rule

Deciphering the pattern that converts the input into an output value in each table and writing it as an expression is all one is expected to do in this set of printable function rule worksheets for high school students.

Graphing Linear Functions Worksheets

Plotting coordinates and graphing functions are the two major learning outcomes of this section of graphing linear equations worksheet printables. The slopes here are expressed as integers and fractions.

(30 Worksheets)

Comparing Linear Functions

Compare two functions presented as tables, graphs and equations in these printable worksheets. Find the slope, x and y-intercepts to complete the table and answer questions based on comparisons.

Transformation of Linear Functions Worksheets

Draw a transformed graph, write a transformed function using horizontal and vertical shifts, stretches, compressions, and reflections too with these transformation of linear functions worksheets.

(25 Worksheets)

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Linear Equation Worksheets (Distributive Property)

Related Topics: More Math Worksheets More Grade 7 Math Lessons Grade 7 Math Worksheets

There are five sets of solving linear equation worksheets:

• Solve linear equations (1-step equations)
• Solve linear equations (2-step equations)
• Solve linear equations (combine like terms)
• Solve linear equations (distributive property & combine like terms)
• Solve linear equations (fractions)

Examples, solutions, videos, and worksheets to help Grade 6 and Grade 7 students learn how to solve linear equations by using the distributive property and combining like terms.

How to solve linear equations (distributive property & combining like terms)?

There are three sets of solving linear equations worksheets using distributive property & combining like terms.

• Solve linear equations (variables on one side, positive coefficients).
• Solve linear equations (variables on one side, negative coefficients).
• Solve linear equations (variables on both sides).

You can follow these steps to solve a linear equation:

• Use the distributive property and combine like terms, if applicable.
• Multiply each term inside the parentheses by the number or variable outside the parentheses.
• Combine the like terms that were created by the distribution.
• Isolate the variable term. Perform inverse operations to move all other terms to the opposite side of the equation.
• Simplify the equation further, if necessary, until you have the variable term isolated on one side of the equation.
• Solve for the variable by performing the necessary operations to isolate it.
• Check your solution by substituting the value you found back into the original equation and verifying that both sides are equal.

For example, 3x + 3(2x + 3) = 54 3x + 6x + 9 = 54 9x + 9 = 54 9x = 54 x = 5

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Linear Equations Worksheets .

More Linear Equation Worksheets (Distributive Property & Combine Like Terms)

Printable (Answers on the second page.) Linear Equations Worksheet #1 (variables on one side, positive coefficients) Linear Equations Worksheet #2 (variables on one side, negative coefficients) Linear Equations Worksheet #3 (variables on both sides)

Online or Generated Combine like Terms (Sprint) Solve by Combining Like Terms Solve Equations with the Variable on Both Sides Solve Equations by Distributive Property Solve Multi-Step Equations 1 Solve Multi-Step Equations 2 Solve Word Problems using Algebra

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Linear Equations

On these printable worksheets, students will practice solving, finding x and y intercepts, and graphing linear equations.

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STW has a variety of algebra and pre-algebra topics, including: evaluating algebraic expressions, algebraic equations (single-step), algebraic equations (two-step), dependent & independent variables, and inequalities.

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Unit 3: Linear equations and functions

About this unit.

When distances, prices, or any other quantity in our world changes at a constant rate, we can use linear functions to model them. Let's learn how different representations, including graphs and equations, of these useful functions reveal characteristics of the situation.

Graphing proportional relationships

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• What is a function? (Opens a modal)
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• Equations vs. functions (Opens a modal)
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Linear models

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Comparing linear functions

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Constructing linear models for real-world relationships

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Recognizing functions

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Linear and nonlinear functions

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How Can Graphing Linear Function Worksheets Help?

Functions worksheets allow students to learn more about the equations that can be linear functions, how to solve the equations, and how to graph linear functions.

What Worksheets are Available?

There are functions worksheets available that review drawing linear functions, drawing slopes, finding slopes, slope-intercept form, and more.

Writing Linear Equations

If a graph or two points are given, it is possible to write out the linear function using information from the graph.

What are the Methods for Writing Linear Equations?

There are a couple of methods for writing linear equations. The first method is to find one point, and the slope on a graph then put the information in an equation in slope-intercept form. Another is to use two points on the graph or points that are given to create the linear equation.

What Do You Need to Write Linear Equations?

To write a linear equation, you need a graph, two points that can be plotted on a graph, or one point along with a slope.

Word Problems for Linear Functions

Word problems that involve linear functions can be solved by looking for the right information, then plotting it on a graph or using the information to create the equation.

What Information Do You Look for in Word Problems?

In a word problem, look for two points that can be plotted on a graph or determine the y-intercept and the slope. If possible, go ahead and plot these on the graph to make it easier to see the function.

How Do You Solve Word Problems?

Solve word problems by finding the information needed then writing the equation. Plot the function if possible, to make sure the answer is correct.

Cazoom Maths Linear Function Worksheets Are Easy to Understand

Linear functions are considered easy to understand because they are just a straight line on a graph. Use the linear function worksheets available to learn how to write, solve, and plot linear functions to get a better understanding of how they work and be able to solve them easily in any format.

1. Columbia University

http://www.columbia.edu/itc/sipa/math/linear.html

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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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Linear Equations worksheets are an essential tool for teachers to help their students master the fundamental concepts of math and algebra. These worksheets provide a variety of problems that challenge students to apply their knowledge and skills in solving linear equations. Teachers can use these worksheets to supplement their lesson plans, provide additional practice for students who may be struggling with the material, or even as a form of assessment to gauge student understanding. With a wide range of topics covered, from simple one-step equations to more complex systems of linear equations, these worksheets cater to students of all grade levels and abilities. By incorporating Linear Equations worksheets into their teaching strategies, educators can ensure that their students are well-equipped to tackle the challenges of math and algebra with confidence.

Quizizz offers a comprehensive collection of Linear Equations worksheets, as well as other math and algebra resources, to support teachers in their quest to provide the best possible education for their students. With Quizizz, teachers can easily create engaging and interactive quizzes that can be used in conjunction with worksheets to reinforce learning and assess student progress. In addition to Linear Equations worksheets, Quizizz also offers a wide range of other math and algebra resources, including lessons, activities, and games, all designed to make learning fun and engaging for students of all grade levels. By utilizing Quizizz in their classrooms, teachers can create a dynamic and interactive learning environment that not only helps students master the concepts of linear equations but also fosters a lifelong love of learning.

Linear Equations Word Problems 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 word problems worksheets have a variety of word problems that help students practice key concepts and build a rock-solid foundation of the concepts.

Benefits of Linear Equations Word Problems Worksheets

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

Linear equations word problems 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.

Download Linear Equations Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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4.1E: Linear Functions (Exercises)

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1. Determine whether the algebraic equation is linear. \(2 x+3 y=7\) 2. Determine whether the algebraic equation is linear. \(6 x^{2}-y=5\) 3. Determine whether the function is increasing or decreasing. \[f(x)=7 x-2 \nonumber\] 4. Determine whether the function is increasing or decreasing. \[g(x)=-x+2 \nonumber\] 5. Given each set of information, find a linear equation that satisfies the given conditions, if possible. Passes through (7,5) and (3,17) 6. Given each set of information, find a linear equation that satisfies the given conditions, if possible. \(x\) -intercept at (6,0) and \(y\) -intercept at (0,10) 7. Find the slope of the line shown in the graph.

Does the following table represent a linear function? If so, find the linear equation that models the data.

12. On June \(1^{\text {st }}\), a company has \(\$ 4,000,000\) profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the company's profit \(n^{\text {th }}\) day after June \(1^{\text {st }}\) ?

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

13. \[\begin{align*} 2 x-6 y &=12 \\[4pt] -x+3 y &=1 \end{align*}\]

14 \[\begin{align*} y &=\frac{1}{3} x-2 \\[4pt] 3 x+y &=-9 \end{align*}\]

For the following exercises, find the \(x\) - and \(y\) - intercepts of the given equation

15. \(7 x+9 y=-63\) 16. \(f(x)=2 x-1\)

For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line \(2 .\) Is each pair of lines parallel, perpendicular, or neither?

• Line 1 : Passes through (5,11) and (10,1)
• Line 2: Passes through (-1,3) and (-5,11)
• Line 1: Passes through (8,-10) and (0,-26)
• Line 2: Passes through (2,5) and (4,4)

19. Write an equation for a line perpendicular to \(f(x)=5 x-1\) and passing through the point (5,20).

20. Find the equation of a line with a \(y\) - intercept of (0,2) and slope \(-\frac{1}{2}\).

23. A car rental company offers two plans for renting a car.

• Plan A: 25 dollars per day and 10 cents per mile
• Plan B: 50 dollars per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

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Linear Function Word Problems Worksheets

What is a Linear Function? Linear functions are those algebraic functions which have a graphical figure of straight lines consisting of unique values on their slopes and y-intercept. In other words, a linear function is an equation which is either a constant or a product of a constant whose highest power is 1. For instance, the perfect example of a linear equation is y=mx+b, while the equation is called the slope-intercept form. This equation meets all the criteria of a linear function, i.e. two variables in the form of x and y, while also having two constants m and b. in the equation you can also see that the highest power of the variables is 1. A series of worksheets and lessons that help students tackle word problems that include functions that can be graphed to form straight lines.

Aligned Standard: HSS-ID.B.6a

• John's Rice Buying Step-by-step Lesson - Create an equation to represent the rate at which John purchases rice.
• Guided Lesson - Nail growth, borrowing bicycles, and a leaky water faucet.
• Guided Lesson Explanation - I somewhat shot through the explanation quickly.
• Practice Worksheet - Baby's hair growth, filling water tanks, and the Earth's temperature.
• Matching Worksheet - Find the graph of each situation that is explained to you.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

You would not imagine how often linear functions are found in everyday problems. These problems will give you a glimpse.

• Homework 1 - Determine the equation and represent the function that defines the cost of corn based on weight.
• Homework 2 - It has been observed that a particular hair's growth is directly proportional to time. When we last measured, the hair was 5 cm and 6.5 cm exactly one week later. If the hair continues to grow at this rate, determine the function that represents the hair's growth and graph it.
• Homework 3 - Aryan wants to borrow a car. He borrows a car from Frieda. The car rental charge is $12 per day plus $0.40 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 100 miles were travelled in two days, how much is the rental cost?

Practice Worksheets

When the graph answers are printed in pure grayscale, they come out great!

• Practice 1 - Henry has been measuring the length of his baby's nail. The first time it was 2 cm long and after one month it was 2.2 cm long. If the nail continues to grow at this rate, determine the function that represents the nail growth and graph it.
• Practice 2 - Denzel is making cups of homemade hot chocolate at the rate of 7 cups in an hour. Create a hypothetical table of values for time and capacity. Determine the equation that represents the function and state it graphically.
• Practice 3 - Goldie has purchased five pounds of copper tubing from the hardware store for $100. Determine the equation and represent the function that defines the cost of copper tubing based on weight.

Math Skill Quizzes

The first quiz will tell you if the kids have the concept. The third quiz will tell you if they mastered the skill.

• Quiz 1 - Gerald has purchased four pairs of sticks from the market for $10. Determine the function that defines the cost of one pair of sticks based on quantity.
• Quiz 2 - Eva writes 3 pages in an hour. Determine the equation that represents the function and state it graphically.
• Quiz 3 - It has been observed that a particular plant's growth is directly proportional to time. When we last measured, the plant was 4 cm and 4.6 cm exactly one week later. If the plant continues to grow at this rate, determine the function that represents the plant's growth and graph it.

Tips for Solving Linear Functions

As we know these algebraic statements can be used to form lines. There is a pretty straight forward strategy you can use when approaching these types of problems and it falls down into three steps. The first step is to swap out the f(x) value and fit into the problem. The next step is to get the variable alone and by itself. While I say this is the next step, it can take you several moves to make it happen. It usually starts with countering the constant value. You can to this by just applying the opposite operation to both sides of the equation. After that you will need to continue through the problem by countering a series of operations the end goal is the get that variable by itself equal to something else. The last step is often to simplify what ever you are left with. This can sometimes be finishing out an exponent, combining a series of operations, or just simplifying a fraction or series of them.

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