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8.ee.c.8. analyze and solve pairs of simultaneous linear equations..

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8.EE.C.8.a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

  • The Intersection of Two Lines

8.EE.C.8.b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, $3x + 2y = 5$ and $3x + 2y = 6$ have no solution because $3x + 2y$ cannot simultaneously be $5$ and $6$.

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8.EE.C.8.c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

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Math  /  8th Grade  /  Unit 6: Systems of Linear Equations

Systems of Linear Equations

Students explore what happens when you consider two linear equations simultaneously, and explore the many rich applications that can be modeled with systems of linear equations in two variables.

Unit Summary

In Unit 6, 8th grade students explore what happens when you consider two linear equations simultaneously. They graph two lines in the same coordinate plane and ask themselves what coordinate points satisfy both of the equations. They consider what it means when two lines never intersect or when they overlap completely. Students learn algebraic methods that can be used to solve systems when graphing is not efficient. Using the structure of the equations in a system, students will determine if systems have one, no, or infinite solutions without solving the system (MP.7). Students also explore the many rich applications that can be modeled with systems of linear equations in two variables (MP.4).

Students will use their knowledge from previous 8th grade units , including work with single linear equations and functions from clusters 8.EE.B and 8.F.B. They will also need to draw on concepts from 6th grade , where they understood solving an equation as a process of answering which values make an equation true. 

In high school , students will continue their work with systems, working with linear, absolute value, quadratic, and exponential functions. They will also graph linear inequalities and consider what the solution of a system of linear inequalities looks like in the coordinate plane.

Pacing: 15 instructional days (11 lessons, 3 flex days, 1 assessment day)

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solve real world and mathematical problems leading to two linear equations in two variables

The following assessments accompany Unit 6.

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Pre-Unit Student Self-Assessment

Have students complete the Mid-Unit Assessment after lesson 6.

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Post-Unit Assessment Answer Key

Post-Unit Student Self-Assessment

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Intellectual Prep

Suggestions for how to prepare to teach this unit

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

Internalization of Standards via the Post-Unit Assessment

  • Standards that each question aligns to
  • Strategies and representations used in daily lessons
  • Relationship to Essential Understandings of unit 
  • Lesson(s) that Assessment points to

Internalization of Trajectory of Unit

  • Read and annotate the Unit Summary.
  • Notice the progression of concepts through the unit using the Lesson Map.

Essential Understandings

  • Connection to Post-Unit Assessment questions
  • Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

  • Read the Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations for the standards relevant to this unit. 

The central mathematical concepts that students will come to understand in this unit

  • A system of linear equations is a set of two or more linear equations that involve the same, related variables.
  • The solution to a system of linear equations represents all of the points that satisfy all of the equations in the system simultaneously. This is seen graphically as the intersecting or overlapping points on the graph and can be verified algebraically by confirming the coordinate point(s) satisfy the equations when they are substituted in.
  • Systems of equations can have one unique solution (intersecting lines), no solution (non-overlapping parallel lines), or infinite solutions (completely overlapping lines).
  • Systems of equations are useful to model real-world situations when more than one equation and more than one variable are involved.

Terms and notation that students learn or use in the unit

elimination (linear combinations)

infinite solutions

no solution

substitution (to solve a system of equations)

system of linear equations

unique solution

To see all the vocabulary for Unit 6, view our 8th Grade Vocabulary Glossary .

Topic A: Analyze & Solve Systems of Equations Graphically

Define a system of linear equations and its solution.

Solve systems of linear equations by graphing. 

8.EE.C.8.A 8.EE.C.8.B

Classify systems of linear equations as having a unique solution, no solutions, or infinite solutions. 

Solve real-world and mathematical problems by graphing systems of linear equations. 

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Topic B: Analyze & Solve Systems of Equations Algebraically

Solve systems of linear equations using substitution when one equation is already solved for a variable.

Solve systems of linear equations using substitution by first solving an equation for a variable. 

 Solve real-world and mathematical problems using linear systems and substitution.

Solve systems of linear equations using elimination (linear combinations) when there is already a zero pair. 

Solve systems of linear equations using elimination (linear combinations) by first creating a zero pair.

Solve real-world and mathematical problems using systems and any method of solution.

8.EE.C.8.B 8.EE.C.8.C

Model and solve real-world problems using systems of equations.

Common Core Standards

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

8.EE.C.8.A — Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8.B — Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

8.EE.C.8.C — Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

6.EE.B.5 — Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.C.7 — Solve linear equations in one variable.

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Future Standards

Standards in future grades or units that connect to the content in this unit

Creating Equations

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Reasoning with Equations and Inequalities

A.REI.C.5 — Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A.REI.C.7 — Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.

A.REI.C.8 — Represent a system of linear equations as a single matrix equation in a vector variable.

A.REI.C.9 — Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP6 — Attend to precision.

CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Linear Relationships

Pythagorean Theorem and Volume

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solve real world and mathematical problems leading to two linear equations in two variables

Common Core State Standards Initiative

Expressions & Equations

Apply and extend previous understandings of arithmetic to algebraic expressions., reason about and solve one-variable equations and inequalities., represent and analyze quantitative relationships between dependent and independent variables., use properties of operations to generate equivalent expressions., solve real-life and mathematical problems using numerical and algebraic expressions and equations., expressions and equations work with radicals and integer exponents., understand the connections between proportional relationships, lines, and linear equations., analyze and solve linear equations and pairs of simultaneous linear equations..

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  • Mathematics Appendix A

solve real world and mathematical problems leading to two linear equations in two variables

Solve real-world and mathematical problems leading to two linear equations in two variables.

Popular tutorials in solve real-world and mathematical problems leading to two linear equations in two variables..

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How Do You Use a System of Linear Equations to Find Coordinates on a Map?

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What's an Example of a Word Problem That Has a System of Linear Equations with Infinite Solutions?

What's an Example of a Word Problem That Has a System of Linear Equations with Infinite Solutions?

How Do You Solve a Word Problem Using the Elimination by Subtraction Method?

How Do You Solve a Word Problem Using the Elimination by Subtraction Method?

Word problems are a great way to see math in action! In this tutorial, you'll see how to write a system of linear equations from the information given in a word problem. Then, you'll see how to solve this system using the elimination method. See this entire process by watching this tutorial!

How Do You Solve a Word Problem Using Two Equations?

How Do You Solve a Word Problem Using Two Equations?

Sometimes word problems describe a system of equations, two equations each with two unknowns. Solving word problems like this one aren't so bad if you know what to do. Check it out with this tutorial!

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Other topics in analyze and solve pairs of simultaneous linear equations. :.

  • Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
  • Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
  • Terms of Use

CCSS.Math.Content.8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables.

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MAFS.8.EE.3.8 Archived Standard

Export to Word

10 Lesson Plans

8 Problem-Solving Tasks

7 Tutorials

7 Formative Assessments

1 Student Center Activity

1 Educational Software / Tool

  • MFAS Formative Assessments 6

Student Resources

  • Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
  • Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
  • Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

  • Assessment Limits : Numbers in items must be rational numbers. Coefficients of equations in standard form must be integers. Items written for MAFS.8.EE.3.8a must include the graph or the equations. Equations in items written for MAFS.8.EE.3.8a must be given in slope-intercept form.
  • Test Item #: Sample Item 1

Use the Add Point tool to plot the solution of the system.

solve real world and mathematical problems leading to two linear equations in two variables

  • Difficulty: N/A
  • Type: GRID: Graphic Response Item Display
  • Test Item #: Sample Item 2

Analyze the system of two equations shown.

How many solutions does the system of equations have?

  • Type: SHT: Selectable Hot Text
  • Test Item #: Sample Item 3

A graph of a system of two equations is shown.

solve real world and mathematical problems leading to two linear equations in two variables

What is the solution of the system?

  • Type: EE: Equation Editor
  • Test Item #: Sample Item 4

solve real world and mathematical problems leading to two linear equations in two variables

What is the approximate solution of the system?

  • Test Item #: Sample Item 5

A system of two equations is shown.

A. Use the Add Arrow tool to graph the two lines. 

B. Drag the palette image to show the solution of the system.

solve real world and mathematical problems leading to two linear equations in two variables

  • Test Item #: Sample Item 6

Company A charges a $25 initial fee and an additional $5 for each hour rented. Company B charges an initial $18 fee and an additional $6 for each hour rented. 

The total cost to rent a bike from Company A can be represented by the equation y=5h+25, where h represents the number of hours rented and y represents the cost, in dollars. 

The total cost to rent a bike from Company B can be represented by the equation y=6h+18, where h represents the number of hours rented and y represents the cost, in dollars. 

For how many hours of rental is the amount charged by the two companies the same? What is the cost, in dollars, of renting the bike for this many hours?

  • Test Item #: Sample Item 7
  • Test Item #: Sample Item 8

Select the number of solutions for each system of two linear equations.

  • Type: MI: Matching Item

Related Courses

Related access points, related resources, educational software / tool.

A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Type: Educational Software / Tool

Formative Assessments

Students are asked to determine the number of solutions of each of four systems of linear equations without solving the systems of equations.

Type: Formative Assessment

Students are asked to solve a system of linear equations by graphing.

Students are asked to identify the solutions of systems of equations from their graphs and justify their answers.

Students are given word problems and asked to write a pair of simultaneous linear equations that could be used to solve them.

Students are asked to solve a word problem by solving a system of linear equations.

Students are asked to solve three systems of linear equations algebraically.

  • Interpret a situation and represent the variables mathematically.
  • Select appropriate mathematical methods to use.
  • Explore the effects of systematically varying the constraints.
  • Interpret and evaluate the data generated and identify the break-even point, checking it for confirmation.
  • Communicate their reasoning clearly.

Lesson Plans

This lesson is designed to introduce solving systems of linear equations in two variables by graphing. Students will find the solutions of systems of linear equations in two variables by graphing "paths" of battleships and paths of launched torpedoes targeting them. The solutions of the systems will represent the intersection of the paths of a targeted ship (modeled by a linear equation) and the path of a torpedo from a battleship (modeled by another linear equation).

Type: Lesson Plan

  • Using substitution to complete a table of values for a linear equation.
  • Identifying a linear equation from a given table of values.
  • Graphing and solving linear equations.

This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the variables mathematically, select appropriate mathematical methods to use, explore the effects of systematically varying the constraints, interpret and evaluate the data generated and identify the break-even point, checking it for confirmation and communicate their reasoning clearly.

Students will graph systems of linear equations in slope-intercept form to find the solution to the system, the point of intersection. Because the lesson builds upon a group activity, the students have an easy flow into the lesson and the progression of the lesson is a smooth transition into solving systems algebraically.

Students will graph two linear functions using pieces of string that intersect and discover what the point of intersection has to do with both functions. It will get tricky when the functions do not intersect, or when they transform into the same equation.

Students will learn to find the solutions to a system of linear equations, by graphing the equations.

This lesson plan introduces the concept of graphing a system of linear equations. Students will use graphing technology to explore the meaning of the solution of a linear system including solutions that correspond to intersecting lines, parallel lines, and coinciding lines. Students will also do graph linear systems by hand.

This lesson unit is intended to help you assess how well students are able to classify solutions to a pair of linear equations by considering their graphical representations. In particular, this unit aims to help you identify and assist students who have difficulties in using substitution to complete a table of values for a linear equation, identifying a linear equation from a given table of values and graphing and solving linear equations.

Students write and solve linear equations from real-life situations. 

Students will work in cooperative groups to demonstrate solving systems of linear equations. They will form lines as a group and see where the point of intersection is.

Problem-Solving Tasks

The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions or as an assessment tool, with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.

Type: Problem-Solving Task

This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns. This progression of tasks helps distinguish between 8th grade and high school expectations related to systems of linear equations.

This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the cell phone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task.

In this task, we are given the graph of two lines including the coordinates of the intersection point and the coordinates of the two vertical intercepts and are asked for the corresponding equations of the lines. It is a very straightforward task that connects graphs and equations and solutions and intersection points.

The task is a modeling problem which ties in to financial decisions faced routinely by businesses, namely the balance between maintaining inventory and raising short-term capital for investment or re-investment in developing the business.

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table (Standard for Mathematical Practice, ).

Students are asked to write equations to model the repair costs of three different companies and determine the conditions for which each company would be least expensive. This task can be used to both assess student understanding of systems of linear equations or to promote discussion and student thinking that would allow for a stronger solidification of these concepts. The solution can be determined in multiple ways, including either a graphical or algebraic approach.

The student is given the equation 5x-2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.

Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

This example demonstrates solving a system of equations algebraically and graphically.

Type: Tutorial

This video demonstrates a system of equations with no solution.

This video shows how to solve a system of equations using the substitution method.

This video demonstrates testing a solution (coordinate pair) for a system of equations

This video demonstrates analyzing solutions to linear systems using a graph.

This video shows how to algebraically analyze a system that has no solutions.

When solving a system of linear equations in x and y with a single solution, we get a unique pair of values for x and y. But what happens when try to solve a system with no solutions or an infinite number of solutions?

MFAS Formative Assessments

Student resources, parent resources.

solve real world and mathematical problems leading to two linear equations in two variables

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2.5.4: Applications Using Linear Models

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Applications Using Linear Models

Suppose a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. Could you write a linear equation to model this situation? Would slope-intercept form, point-slope form, or standard form be easiest to use?

Applying Linear Models

Modeling linear relationships can help solve real-world applications. Consider the example situations below, and note how different problem-solving methods may be used in each.

  • Nadia has $200 in her savings account. She gets a job that pays $7.50 per hour and she deposits all her earnings in her savings account. Write the equation describing this problem in slope-intercept form. How many hours would Nadia need to work to have $500 in her account?

solve real world and mathematical problems leading to two linear equations in two variables

Begin by defining the variables:

y= amount of money in Nadia’s savings account

x= number of hours

The y-intercept ($200) and the slope of the equation ($7.501 hour) are given.

We are told that Nadia has $200 in her savings account, so b=200.

We are told that Nadia has a job that pays $7.50 per hour, so m=7.50.

By substituting these values into slope–intercept form, y=mx+b, we obtain y=7.5x+200.

To answer the question, substitute $500 for the value of y and solve.

500=7.5x+200

Nadia must work 40 hours if she is to have $500 in her account.

  • Marciel rented a moving truck for the day. Marciel remembers only that the rental truck company charges $40 per day and some amount of cents per mile. Marciel drives 46 miles and the final amount of the bill (before tax) is $63. What is the amount per mile the truck rental company charges? Write an equation in point-slope form that describes this situation. How much would it cost to rent this truck if Marciel drove 220 miles?

solve real world and mathematical problems leading to two linear equations in two variables

Define the variables: x= distance in miles; y= cost of the rental truck in dollars. There are two ordered pairs: (0, 40) and (46, 63).

Step 1: Begin by finding the slope: 63−4046−0=2346=12

Step 2: Substitute the slope for m and one of the coordinates for (x1,y1).

y−40=12(x−0)

To find out how much will it cost to rent the truck for 220 miles, substitute 220 for the variable x.

y−40=1/2(220−0)

y−40=0.5(220)

  • Nimitha buys fruit at her local farmer’s market. This Saturday, oranges cost $2 per pound and cherries cost $3 per pound. She has $12 to spend on fruit. Write an equation in standard form that describes this situation. If she buys 4 pounds of oranges, how many pounds of cherries can she buy?

solve real world and mathematical problems leading to two linear equations in two variables

Define the variables: x= pounds of oranges and y= pounds of cherries.

The equation that describes this situation is: 2x+3y=12

If she buys 4 pounds of oranges, we substitute x=4 into the equation and solve for y.

3y=12−8

Nimitha can buy 1(1/3) pounds of cherries.

Example 2.5.4.1

Earlier, you were told that a movie rental service charges a fixed fee per month and also charges $3.00 per movie rented. Last month you rented 8 movies and your monthly bill was $30.00. What linear equation would model this situation?

In this example, you are given the slope of the line that would represent this situation: 3 (because each rental costs $3.00). You are also given the point (8, 30) because when you rent 8 movies, your bill is $30.00. So, you have the slope and a point. This means that the best form to use to write an equation is point-slope form.

To write the equation, first define the variables: x= number of movies rented; y= the monthly bill in dollars. The slope is 3 and one ordered pair is (8, 30).

Since you have the slope, substitute the slope for m and the coordinate for (x1,y1) into the point-slope form equation:

y−30=3(x−8)

You can rewrite this in slope-intercept form by using the Distributive Property and the Addition Property of Equality:

y−30=3x−24

So the equation that models this situation is y−30=3(x−8) or y=3x+6.

Example 2.5.4.2

A stalk of bamboo of the family Phyllostachys nigra grows at steady rate of 12 inches per day and achieves its full height of 720 inches in 60 days. Write the equation describing this problem in slope-intercept form. How tall is the bamboo 12 days after it started growing?

Define the variables.

y= the height of the bamboo plant in inches

x= number of days

The problem gives the slope of the equation and a point on the line.

The bamboo grows at a rate of 12 inches per day, so m=12.

We are told that the plant grows to 720 inches in 60 days, so we have the point (60, 720).

Start with the slope-intercept form of the line. y=mx+b

Substitute 12 for the slope. y=12x+b

Substitute the point (60,720). 720=12(60)+b ⇒b=0

Substitute the value of b back into the equation. y=12x

To answer the question, substitute the value x=12 to obtain y=12(12)=144 inches.

The bamboo is 144 inches 12 days after it starts growing.

Example 2.5.4.3

Jethro skateboards part of the way to school and walks for the rest of the way. He can skateboard at 7 miles per hour and he can walk at 3 miles per hour. The distance to school is 6 miles. Write an equation in standard form that describes this situation. If Jethro skateboards for 12 of an hour, how long does he need to walk to get to school?

solve real world and mathematical problems leading to two linear equations in two variables

Define the variables: x= hours Jethro skateboards and y= hours Jethro walks.

The equation that describes this situation is 7x+3y=6.

If Jethro skateboards 12 of an hour, we substitute x=0.5 into the equation and solve for y.

7(0.5)+3y=6

3y=6−3.5

Jethro must walk 5/6 of an hour to get to school.

  • To buy a car, Andrew puts in a down payment of $1500 and pays $350 per month in installments. Write an equation describing this problem in slope-intercept form. How much money has Andrew paid at the end of one year?
  • Anne transplants a rose seedling in her garden. She wants to track the growth of the rose, so she measures its height every week. In the third week, she finds that the rose is 10 inches tall and in the eleventh week she finds that the rose is 14 inches tall. Assuming the rose grows linearly with time, write an equation describing this problem in slope-intercept form. What was the height of the rose when Anne planted it?
  • Ravi hangs from a giant exercise spring whose length is 5 m. When his child Nimi hangs from the spring, his length is 2 m. Ravi weighs 160 lbs. and Nimi weighs 40 lbs. Write the equation for this problem in slope-intercept form. What should we expect the length of the spring to be when his wife Amardeep, who weighs 140 lbs., hangs from it?
  • Petra is testing a bungee cord. She ties one end of the bungee cord to the top of a bridge and to the other end she ties different weights. She then measures how far the bungee stretches. She finds that for a weight of 100 lbs., the bungee stretches to 265 feet and for a weight of 120 lbs., the bungee stretches to 275 feet. Physics tells us that in a certain range of values, including the ones given here, the amount of stretch is a linear function of the weight. Write the equation describing this problem in slope-intercept form. What should we expect the stretched length of the cord to be for a weight of 150 lbs?
  • Nadia is placing different weights on a spring and measuring the length of the stretched spring. She finds that for a 100 gram weight the length of the stretched spring is 20 cm and for a 300 gram weight the length of the stretched spring is 25 cm. Write an equation in point-slope form that describes this situation. What is the unstretched length of the spring?
  • Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to a depth of 50 feet. Write an equation in point-slope form that describes this situation. What was the submarine’s depth five minutes after it started surfacing?
  • Anne got a job selling window shades. She receives a monthly base salary and a $6 commission for each window shade she sells. At the end of the month, she adds up her sales and she figures out that she sold 200 window shades and made $2500. Write an equation in point-slope form that describes this situation. How much is Anne’s monthly base salary?
  • The farmer’s market sells tomatoes and corn. Tomatoes are selling for $1.29 per pound and corn is selling for $3.25 per pound. If you buy 6 pounds of tomatoes, how many pounds of corn can you buy if your total spending cash is $11.61?
  • The local church is hosting a Friday night fish fry for Lent. They sell a fried fish dinner for $7.50 and a baked fish dinner for $8.25. The church sold 130 fried fish dinners and took in $2,336.25. How many baked fish dinners were sold?
  • Andrew has two part-time jobs. One pays $6 per hour and the other pays $10 per hour. He wants to make $366 per week. Write an equation in standard form that describes this situation. If he is only allowed to work 15 hours per week at the $10 per hour job, how many hours does he need to work per week at his $6 per hour job in order to achieve his goal?
  • Anne invests money in two accounts. One account returns 5% annual interest and the other returns 7% annual interest. In order not to incur a tax penalty, she can make no more than $400 in interest per year. Write an equation in standard form that describes this problem. If she invests $5000 in the 5% interest account, how much money does she need to invest in the other account?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.6.

Additional Resources

PLIX: Play, Learn, Interact, eXplore: The Perfect Lemonade 1

Activity: Applications Using Linear Models Discussion Questions

Practice: Applications Using Linear Models

Real World Application: Tracking the Storm

OML Search

Systems of Equations (Elimination)

Suggested learning targets.

  • I can identify the solution(s) to a system of two linear equations in two variables as the point(s) of intersection of their graphs.
  • I can describe the point(s) of intersection between two lines as the points that satisfy both equations simultaneously.
  • I can define "inspection."
  • I can solve a system of two equations (linear) in two unknowns algebraically.
  • I can identify cases in which a system of two equations in two unknowns has no solution.
  • I can identify cases in which a system of two equations in two unknowns has an infinite number of solutions.
  • I can solve simple cases of systems of two linear equations in two variables by inspection.
  • I can estimate the point(s) of intersection for a system of two equations in two unknowns by graphing the equations.
  • I can represent real-world and mathematical problems leading to two linear equations in two variables.

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  • 5.9 Systems of Linear Equations in Two Variables
  • Introduction
  • 1.1 Basic Set Concepts
  • 1.2 Subsets
  • 1.3 Understanding Venn Diagrams
  • 1.4 Set Operations with Two Sets
  • 1.5 Set Operations with Three Sets
  • Key Concepts
  • Formula Review
  • Chapter Review
  • Chapter Test
  • 2.1 Statements and Quantifiers
  • 2.2 Compound Statements
  • 2.3 Constructing Truth Tables
  • 2.4 Truth Tables for the Conditional and Biconditional
  • 2.5 Equivalent Statements
  • 2.6 De Morgan’s Laws
  • 2.7 Logical Arguments
  • 3.1 Prime and Composite Numbers
  • 3.2 The Integers
  • 3.3 Order of Operations
  • 3.4 Rational Numbers
  • 3.5 Irrational Numbers
  • 3.6 Real Numbers
  • 3.7 Clock Arithmetic
  • 3.8 Exponents
  • 3.9 Scientific Notation
  • 3.10 Arithmetic Sequences
  • 3.11 Geometric Sequences
  • 4.1 Hindu-Arabic Positional System
  • 4.2 Early Numeration Systems
  • 4.3 Converting with Base Systems
  • 4.4 Addition and Subtraction in Base Systems
  • 4.5 Multiplication and Division in Base Systems
  • 5.1 Algebraic Expressions
  • 5.2 Linear Equations in One Variable with Applications
  • 5.3 Linear Inequalities in One Variable with Applications
  • 5.4 Ratios and Proportions
  • 5.5 Graphing Linear Equations and Inequalities
  • 5.6 Quadratic Equations with Two Variables with Applications
  • 5.7 Functions
  • 5.8 Graphing Functions
  • 5.10 Systems of Linear Inequalities in Two Variables
  • 5.11 Linear Programming
  • 6.1 Understanding Percent
  • 6.2 Discounts, Markups, and Sales Tax
  • 6.3 Simple Interest
  • 6.4 Compound Interest
  • 6.5 Making a Personal Budget
  • 6.6 Methods of Savings
  • 6.7 Investments
  • 6.8 The Basics of Loans
  • 6.9 Understanding Student Loans
  • 6.10 Credit Cards
  • 6.11 Buying or Leasing a Car
  • 6.12 Renting and Homeownership
  • 6.13 Income Tax
  • 7.1 The Multiplication Rule for Counting
  • 7.2 Permutations
  • 7.3 Combinations
  • 7.4 Tree Diagrams, Tables, and Outcomes
  • 7.5 Basic Concepts of Probability
  • 7.6 Probability with Permutations and Combinations
  • 7.7 What Are the Odds?
  • 7.8 The Addition Rule for Probability
  • 7.9 Conditional Probability and the Multiplication Rule
  • 7.10 The Binomial Distribution
  • 7.11 Expected Value
  • 8.1 Gathering and Organizing Data
  • 8.2 Visualizing Data
  • 8.3 Mean, Median and Mode
  • 8.4 Range and Standard Deviation
  • 8.5 Percentiles
  • 8.6 The Normal Distribution
  • 8.7 Applications of the Normal Distribution
  • 8.8 Scatter Plots, Correlation, and Regression Lines
  • 9.1 The Metric System
  • 9.2 Measuring Area
  • 9.3 Measuring Volume
  • 9.4 Measuring Weight
  • 9.5 Measuring Temperature
  • 10.1 Points, Lines, and Planes
  • 10.2 Angles
  • 10.3 Triangles
  • 10.4 Polygons, Perimeter, and Circumference
  • 10.5 Tessellations
  • 10.7 Volume and Surface Area
  • 10.8 Right Triangle Trigonometry
  • 11.1 Voting Methods
  • 11.2 Fairness in Voting Methods
  • 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
  • 11.4 Apportionment Methods
  • 11.5 Fairness in Apportionment Methods
  • 12.1 Graph Basics
  • 12.2 Graph Structures
  • 12.3 Comparing Graphs
  • 12.4 Navigating Graphs
  • 12.5 Euler Circuits
  • 12.6 Euler Trails
  • 12.7 Hamilton Cycles
  • 12.8 Hamilton Paths
  • 12.9 Traveling Salesperson Problem
  • 12.10 Trees
  • 13.1 Math and Art
  • 13.2 Math and the Environment
  • 13.3 Math and Medicine
  • 13.4 Math and Music
  • 13.5 Math and Sports
  • A | Co-Req Appendix: Integer Powers of 10

Learning Objectives

After completing this section, you should be able to:

  • Determine and show whether an ordered pair is a solution to a system of equations.
  • Solve systems of linear equations using graphical methods.
  • Solve systems of linear equations using substitution.
  • Solve systems of linear equations using elimination.
  • Identify systems with no solution or infinitely many solutions.
  • Solve applications of systems of linear equations.

In this section, we will learn how to solve systems of linear equations in two variables. There are several real-world scenarios that can be represented by systems of linear equalities. Suppose two friends, Andrea and Bart, go shopping at a farmers market to buy some vegetables. Andrea buys 2 tomatoes and 4 cucumbers and spends $2.00. Bart buys 4 tomatoes and 5 cucumbers and spends $2.95. What is the price of each vegetable?

Determining If an Ordered Pair Is a Solution to a System of Equations

When we solved linear equations in Linear Equations in One Variable with Applications and Linear Inequalities in One Variable with Applications , we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations .

In this section, we will focus our work on systems of two linear equations in two unknowns (variables) and applications of systems of linear equations. An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

A linear equation in two variables, such as 2 x + y = 7 2 x + y = 7 , has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line. To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x x , y y ) that make both equations true. These are called the solutions of a system of equations .

To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Example 5.81

Determining whether an ordered pair is a solution to the system.

Determine whether the ordered pair is a solution to the system.

  • ( − 2 , − 1 ) ( − 2 , − 1 )
  • ( − 4 , − 3 ) ( − 4 , − 3 )
  • { x − y = − 1 2 x − y = − 5 { x − y = − 1 2 x − y = − 5 We substitute x = − 2 x = − 2 and y = − 1 y = − 1 into both equations. x − y = − 1 − 2 − ( − 1 ) = ? − 1 − 1 = − 1 ✓ 2 x − y = − 5 2 ( − 2 ) - ( − 1 ) = ? − 5 - 3 ≠ − 5 x − y = − 1 − 2 − ( − 1 ) = ? − 1 − 1 = − 1 ✓ 2 x − y = − 5 2 ( − 2 ) - ( − 1 ) = ? − 5 - 3 ≠ − 5 ( − 2 , − 1 ) ( − 2 , − 1 ) does not make both equations true. ( − 2 , − 1 ) ( − 2 , − 1 ) is not a solution.
  • We substitute x = − 4 x = − 4 and y = − 3 y = − 3 into both equations. x − y = − 1 − 4 − ( − 3 ) = ? − 1 − 1 = − 1 ✓ 2 x − y = − 5 2 • ( − 4 ) − ( − 3 ) = ? − 5 − 5 = − 5 ✓ x − y = − 1 − 4 − ( − 3 ) = ? − 1 − 1 = − 1 ✓ 2 x − y = − 5 2 • ( − 4 ) − ( − 3 ) = ? − 5 − 5 = − 5 ✓ ( − 4 , − 3 ) ( − 4 , − 3 ) makes both equations true. ( − 4 , − 3 ) ( − 4 , − 3 ) is a solution.

Your Turn 5.81

Example 5.82.

Determine whether the ordered pair is a solution to the system

  • ( − 4 , − 5 ) ( − 4 , − 5 )
  • ( − 4 , 5 ) ( − 4 , 5 )
  • { y = 3 2 x + 1 2 x − 3 y = 7 { y = 3 2 x + 1 2 x − 3 y = 7 Substitute − 4 − 4 for x x and − 5 − 5 for y y into both equations. − 5 = ? 3 2 ( − 4 ) + 1 − 5 = ? 3 ( − 2 ) + 1 − 5 = ? − 6 + 1 − 5 = − 5 ✓ 2 ( − 4 ) − 3 ( − 5 ) = ? 7 ( − 8 ) − ( − 15 ) = ? 7 − 8 + 15 = ? 7 7 = 7 ✓ − 5 = ? 3 2 ( − 4 ) + 1 − 5 = ? 3 ( − 2 ) + 1 − 5 = ? − 6 + 1 − 5 = − 5 ✓ 2 ( − 4 ) − 3 ( − 5 ) = ? 7 ( − 8 ) − ( − 15 ) = ? 7 − 8 + 15 = ? 7 7 = 7 ✓ ( − 4 , − 5 ) ( − 4 , − 5 ) is a solution.
  • { y = 3 2 x + 1 2 x − 3 y = 7 { y = 3 2 x + 1 2 x − 3 y = 7 Substitute − 4 − 4 for x x and 5 5 for y y into both equations. 5 = ? 3 2 ( − 4 ) + 1 5 = ? 3 ( − 2 ) + 1 5 = ? − 6 + 1 5 ≠ − 5 2 ( − 4 ) − 3 ( 5 ) = ? 7 ( − 8 ) − ( 15 ) = ? 7 − 8 − 15 = ? 7 − 23 ≠ 7 5 = ? 3 2 ( − 4 ) + 1 5 = ? 3 ( − 2 ) + 1 5 = ? − 6 + 1 5 ≠ − 5 2 ( − 4 ) − 3 ( 5 ) = ? 7 ( − 8 ) − ( 15 ) = ? 7 − 8 − 15 = ? 7 − 23 ≠ 7 ( − 4 , 5 ) ( − 4 , 5 ) is not a solution.

Your Turn 5.82

Solving systems of linear equations using graphical methods.

We will use three methods to solve a system of linear equations. The first method we will use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what points the lines have in common, we will find the solution to the system.

Most linear equations in one variable have one solution; but for some equations called contradictions , there are no solutions, and for other equations called identities , all numbers are solutions. Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure 5.83 .

Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end you will decide which method was the most convenient way to solve this system.

The steps to use to solve a system of linear equations by graphing are shown here.

Step 1: Graph the first equation.

Step 2: Graph the second equation on the same rectangular coordinate system.

Step 3: Determine whether the lines intersect, are parallel, or are the same line.

Step 4: Identify the solution to the system.

If the lines intersect, identify the point of intersection. This is the solution to the system.

If the lines are parallel, the system has no solution.

If the lines are the same, the system has an infinite number of solutions.

Step 5: Check the solution in both equations.

Example 5.83

Solving a system of linear equations by graphing.

Solve this system of linear equations by graphing.

Your Turn 5.83

Solving systems of linear equations using substitution.

We will now solve systems of linear equations by the substitution method. We will use the same system we used for graphing.

We will first solve one of the equations for either x x or y 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. This process is summarized here:

Step 1: Solve one of the equations for either variable.

Step 2: Substitute the expression from Step 1 into the other equation.

Step 3: Solve the resulting equation.

Step 4: Substitute the solution in Step 3 into either of the original equations to find the other variable.

Step 5: Write the solution as an ordered pair.

Step 6: Check that the ordered pair is a solution to both original equations.

Example 5.84

Solving a system of linear equations using substitution.

Solve this system of linear equations by substitution:

Your Turn 5.84

Solving systems of linear equations using elimination.

We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small, and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.

The third method of solving systems of linear equations is called the elimination method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we’ll do with the elimination method, too, but we’ll have a different way to get there.

The elimination method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal. For any expressions a a , b b , c c , and d d :

if a = b a = b

and c = d c = d

then a + c = b + d a + c = b + d .

To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable. Notice how that works when we add these two equations together:

The y y ’s add to zero and we have one equation with one variable. Let us try another one:

This time we do not see a variable that can be immediately eliminated if we add the equations. But if we multiply the first equation by − 2 − 2 , we will make the coefficients of x x opposites. We must multiply every term on both sides of the equation by − 2 − 2 .

Then rewrite the system of equations.

Now we see that the coefficients of the x x terms are opposites, so x x will be eliminated when we add these two equations.

Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations. Here’s a summary of using the elimination method:

Step 1: Write both equations in standard form. If any coefficients are fractions, clear them.

Step 2: Make the coefficients of one variable opposites.

Decide which variable you will eliminate.

Multiply one or both equations so that the coefficients of that variable are opposites.

Step 3: Add the equations resulting from Step 2 to eliminate one variable.

Step 4: Solve for the remaining variable.

Step 5: Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.

Step 6: Write the solution as an ordered pair.

Step 7: Check that the ordered pair is a solution to both original equations.

Example 5.85

Solving a system of linear equations using elimination.

Solve this system of linear equations by elimination:

Your Turn 5.85

Identifying systems with no solution or infinitely many solutions.

In all the systems of linear equations so far, the lines intersected, and the solution was one point. In Example 5.86 and Example 5.87 , we will look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

Example 5.86

Solving a system of linear equations with no solution.

Solve the system by a method of your choice:

Let us solve the system of linear equations by graphing.

To graph the first equation, we will use its slope and y y -intercept.

To graph the second equation, we will use the intercepts.

Graph the lines ( Figure 5.84 ).

Determine the points of intersection. The lines are parallel. Since no point is on both lines, there is no ordered pair that makes both equations true. There is no solution to this system.

Your Turn 5.86

Example 5.87, solving a system of linear equations with infinite solutions.

{ y = 2 x − 3 − 6 x + 3 y = − 9 { y = 2 x − 3 − 6 x + 3 y = − 9

Find the slope and y y -intercept of the first equation.

y = 2 x − 3 m = 2 b = − 3 y = 2 x − 3 m = 2 b = − 3

Find the intercepts of the second equation.

− 6 x + 3 y = − 9 − 6 x + 3 y = − 9

Graph the lines ( Figure 5.85 ).

The lines are the same! Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true.

There are infinitely many solutions to this system.

Your Turn 5.87

In the previous example, if you write the second equation in slope-intercept form, you may recognize that the equations have the same slope and same y y -intercept. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system. We say the two lines are coincident. Coincident lines have the same slope and same y y -intercept. A system of equations that has at least one solution is called a consistent system . A system with parallel lines has no solution. We call a system of equations like this an inconsistent system . It has no solution.

We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent, they each have their own set of solutions. Intersecting lines and parallel lines are independent. If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines. Let us sum this up by looking at the graphs of the three types of systems. See Figure 5.86 and the table that follows

WORK IT OUT

Using matrices and cramer’s rule to solve systems of linear equations.

An m m by n n matrix is an array with m m rows and n n columns, where each item in the matrix is a number. Matrices are used for many things, but one thing they can be used for is to represent systems of linear equations. For example, the system of linear equations

can be represented by the following matrix:

To use Cramer’s Rule, you need to be able to take the determinant of a matrix. The determinant of a 2 by 2 matrix A A , denoted | A | | A | , is

For example, the determinant of the matrix | 2 1 3 − 2 | = ( 2 × − 2 ) − ( 3 × 1 ) = − 4 − 3 = − 7. | 2 1 3 − 2 | = ( 2 × − 2 ) − ( 3 × 1 ) = − 4 − 3 = − 7.

Cramer’s Rule involves taking three determinants:

  • The determinant of the first two columns, denoted | D | | D | ;
  • The determinant of the first column and the third column, denoted | D y | | D y | ;
  • The determinant of the third column and the first column, denoted | D x | | D x | .

Going back to the original matrix [ 2 1 7 1 − 2 6 ] [ 2 1 7 1 − 2 6 ]

Now Cramer’s Rule for the solution of the system will be:

Putting in the values for these determinants, we have x = − 20 − 5 = 4 ; y = 5 − 5 = − 1. x = − 20 − 5 = 4 ; y = 5 − 5 = − 1. The solution to the system is the ordered pair ( 4 , − 1 ) ( 4 , − 1 ) .

Solving Applications of Systems of Linear Equations

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we will first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.

Step 1 : Read the problem. Make sure all the words and ideas are understood.

Step 2: Identify what we are looking for.

Step 3: Name what we are looking for. Choose variables to represent those quantities.

Step 4: Translate into a system of equations.

Step 5: Solve the system of equations using good algebra techniques.

Step 6: Check the answer in the problem and make sure it makes sense.

Step 7: Answer the question with a complete sentence.

Example 5.88

Applying system to a real-world application.

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

Step 1: Read the problem.

We are looking for the number of training sessions that would make the pay equal.

Step 3: Name what we are looking for.

Let s = Heather’s salary s = Heather’s salary , and n = the number of training sessions n = the number of training sessions

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.

The system is shown.

Step 5: Solve the system of equations.

We will use substitution.

Substitute 25,000 + 15 n 25,000 + 15 n for s s in the second equation

Solve for n n .

Step 6: Check the answer.

Are 600 training sessions a year reasonable?

Are the two options equal when n = 600 n = 600 ?

Substitute into each equation.

s = 25 , 000 + 15 ( 600 ) = 34 , 000 s = 10 , 000 + 40 ( 600 ) = 34 , 000 s = 25 , 000 + 15 ( 600 ) = 34 , 000 s = 10 , 000 + 40 ( 600 ) = 34 , 000

Step 7: Answer the question.

The salary options would be equal for 600 training sessions.

Your Turn 5.88

Practice with Solving Applications of Systems of Equations

Applications of Systems of Linear Equations

Check Your Understanding

Section 5.9 exercises.

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Linear Equations in Two Variables

A linear equation in two variables is an equation in which two variables have the exponent 1. A system of equations with two variables has a unique solution, no solutions, or infinitely many solutions. A linear system of equations may have 'n' number of variables. An important thing to keep in mind while solving linear equations with n number of variables is that there must be n equations to solve and determine the value of variables.

Linear equations in two variables are the algebraic equations which are of the form (or can be converted to the form) y = mx + b, where m is the slope and b is the y-intercept . They are the equations of the first order. For example, y = 2x + 3 and 2y = 4x + 9 are two-variable linear equations.

What are Linear Equations in Two Variables?

The linear equations in two variables are the equations in which each of the two variables is of the highest order ( exponent ) of 1 and may have one, none, or infinitely many solutions. The standard form of a two-variable linear equation is ax + by + c = 0 where x and y are the two variables. The solutions can also be written in ordered pairs like (x, y). The graphical representation of the pairs of linear equations in two variables includes two straight lines which could be:

  • intersecting lines
  • parallel lines or
  • coincident lines .

Forms of Linear Equations in Two Variables

A linear equation in two variables can be in different forms like standard form , intercept form and point-slope form . For example, the same equation 2x + 3y=9 can be represented in each of the forms like 2x + 3y - 9=0 (standard form), y = (-2/3)x + 3 ( slope-intercept form ), and y - 5/3 = -2/3(x + (-2)) (point-slope form). Look at the image given below showing all these three forms of representing linear equations in two variables with examples.

Linear Equations in Two Variables are of three forms standard form, slope intercept form and the point slope form. Each form is provided with an example

The system of equations means the collection of equations and they are also referred to as simultaneous linear equations . We will learn how to solve pair of linear equations in two variables using different methods.

Solving Pairs of Linear Equations in Two Variables

There are five methods to solve pairs of linear equations in two variables as shown below:

  • Graphical Method
  • Substitution Method
  • Cross Multiplication Method
  • Elimination Method

Determinant Method

Graphical method for solving linear equations in two variables.

The steps to solve linear equations in two variables graphically are given below:

  • Step 1 : To solve a system of two equations in two variables graphically , we graph each equation. To know how, click here or follow steps 2 and 3 below.
  • Step 2 : To graph an equation manually, first convert it to the form y = mx+b by solving the equation for y.
  • Step 3 : Start putting the values of x as 0, 1, 2, and so on and find the corresponding values of y, or vice-versa.
  • Step 4 : Identify the point where both lines meet.
  • Step 5 : The point of intersection is the solution of the given system.

Example: Find the solution of the following system of equations graphically.

Solution: We will graph them and see whether they intersect at a point. As you can see below, both lines meet at (1, 2). Thus, the solution of the given system of linear equations is x=1 and y=2.

Graph of pair of Linear Equations in Two Variables has two lines that meet at the point with coordinates 1 comma 2.

But both lines may not intersect always. Sometimes they may be parallel. In that case, the pairs of linear equations in two variables have no solution. In some other cases, both lines coincide with each other. In that case, each point on that line is a solution of the given system and hence the given system has an infinite number of solutions.

Consistent and Inconsistent System of Linear Equations:

  • If the system has a solution, then it is said to be consistent;
  • otherwise, it is said to be inconsistent.

Independent and Dependent System of Linear Equations:

  • If the system has a unique solution, then it is independent.
  • If it has an infinite number of solutions, then it is dependent. It means that one variable depends on the other.

Consider a system of two linear equations: a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. Here we can understand when a linear system with two variables is consistent/inconsistent and independent/dependent.

Types of Solutions of linear equations in two variables, consistent and inconsistent, dependent and independent

Method of Substitution

To solve a system of two linear equations in two variables using the substitution method , we have to use the steps given below:

  • Step 1: Solve one of the equations for one variable.
  • Step 2: Substitute this in the other equation to get an equation in terms of a single variable.
  • Step 3: Solve it for the variable.
  • Step 4: Substitute it in any of the equations to get the value of another variable.

Example: Solve the following system of equations using the substitution method. x+2y-7=0 2x-5y+13=0

Solution: Let us solve the equation, x+2y-7=0 for y: x+2y-7=0 ⇒2y=7-x ⇒ y=(7-x)/2

Substitute this in the equation, 2x-5y+13=0:

2x-5y+13=0 ⇒ 2x-5((7-x)/2)+13=0 ⇒ 2x-(35/2)+(5x/2)+13=0 ⇒ 2x + (5x/2) = 35/2 - 13 ⇒ 9x/2 = 9/2 ⇒ x=1

Substitute x=1 this in the equation y=(7-x)/2:

y=(7-1)/2 = 3

Therefore, the solution of the given system is x=1 and y=3.

Consider a system of linear equations: a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0.

To solve this using the cross multiplication method , we first write the coefficients of each of x and y and constants as follows:

Cross Multiplication Method to Solve Linear Equations in Two Variables

Here, the arrows indicate that those coefficients have to be multiplied. Now we write the following equation by cross-multiplying and subtracting the products. \(\dfrac{x}{b_{1} c_{2}-b_{2} c_{1}}=\dfrac{y}{c_{1} a_{2}-c_{2} a_{1}}=\dfrac{1}{a_{1} b_{2}-a_{2} b_{1}}\)

From this equation, we get two equations:

\(\begin{align} \dfrac{x}{b_{1} c_{2}-b_{2} c_{1}}&=\dfrac{1}{a_{1} b_{2}-a_{2} b_{1}} \\[0.2cm] \dfrac{y}{c_{1} a_{2}-c_{2} a_{1}}&=\dfrac{1}{a_{1} b_{2}-a_{2} b_{1}} \end{align}\)

Solving each of these for x and y, the solution of the given system is:

\(\begin{align} x&=\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\[0.2cm] y&=\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}} \end{align}\)

Method of Elimination

To solve a system of linear equations in two variables using the elimination method , we will use the steps given below:

  • Step 1: Arrange the equations in the standard form: ax+by+c=0 or ax+by=c.
  • Step 2: Check if adding or subtracting the equations would result in the cancellation of a variable.
  • Step 3: If not, multiply one or both equations by either the coefficient of x or y such that their addition or subtraction would result in the cancellation of any one of the variables.
  • Step 4: Solve the resulting single variable equation.
  • Step 5: Substitute it in any of the given equations to get the value of another variable.

Example: Solve the following system of equations using the elimination method. 2x+3y-11=0 3x+2y-9=0

Adding or subtracting these two equations would not result in the cancellation of any variable. Let us aim at the cancellation of x. The coefficients of x in both equations are 2 and 3. Their LCM is 6. We will make the coefficients of x in both equations 6 and -6 such that the x terms get canceled when we add the equations.

3 × (2x+3y-11=0) ⇒ 6x+9y-33=0 -2 × (3x+2y-9=0) ⇒ -6x-4y+18=0

Now we will add these two equations: 6x+9y-33=0 -6x-4y+18=0

On adding both the above equations we get, ⇒ 5y-15=0 ⇒ 5y=15 ⇒ y=3

Substitute this in one of the given two equations and solve the resultant variable for x. 2x+3y-11=0 ⇒ 2x+3(3)-11=0 ⇒ 2x+9-11=0 ⇒ 2x=2 ⇒ x=1

Therefore, the solution of the given system of equations is x=1 and y=3.

The determinant of a 2 × 2 matrix is obtained by cross-multiplying elements starting from the top left corner and subtracting the products.

Determinant Method to Solve Linear Equations in Two Variables

Consider a system of linear equations in two variables: a 1 x + b 1 y = c 1 and a 2 x + b 2 y = c 2 . To solve them using the determinants method (which is also known as Crammer's Rule ), follow the steps given below:

  • Step 1: We first find the determinant formed by the coefficients of x and y and label it Δ. Δ = \(\left|\begin{array}{ll}a_1 & b_1 \\a_2 & b_2\end{array}\right| = a_1 b_2 - a_2b_1\)
  • Step 2: Then we find the determinant Δ x which is obtained by replacing the first column of Δ with constants. Δ x = \(\left|\begin{array}{ll}c_1 & b_1 \\c_2 & b_2\end{array}\right| = c_1 b_2 - c_2b_1\)
  • Step 3: We then find the determinant Δ y which is obtained by replacing the second column of Δ with constants. Δ y = \(\left|\begin{array}{ll}a_1 & c_1 \\a_2 & c_2\end{array}\right| = a_1 c_2 - a_2c_1\)

Now, the solution of the given system of linear equations is obtained by the formulas:

  • x = Δ x / Δ
  • y = Δ y / Δ

Important Points on Linear Equations with Two Variables:

  • A linear equation in two variables is of the form ax + by + c = 0, where x and y are variables; and a, b, and c are real numbers.
  • A pair of linear equations are of the form a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 and its solution is a pair of values (x, y) that satisfy both equations.
  • To solve linear equations in two variables, we must have at least two equations.
  • A linear equation in two variables has infinitely many solutions.

Tricks and Tips:

While solving the equations using either the substitution method or the elimination method:

  • If we get an equation that is true (i.e., something like 0 = 0, -1 = -1, etc), then it means that the system has an infinite number of solutions.
  • If we get an equation that is false (i.e., something like 0 = 2, 3 = -1, etc), then it means that the system has no solution.

☛Related Topics:

  • Solving Linear Equations Calculator
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Linear Equations in Two Variables Examples

Example 1: The sum of the digits of a two-digit number is 8. When the digits are reversed, the number is increased by 18. Find the number.

Solution: Let us assume that x and y are the tens digit and the ones digit of the required number. Then the number is 10x+y.

And the number when the digits are reversed is 10y+x.

The question says, "The sum of the digits of a two-digit number is 8".

So from this, we get a linear equation in two variables: x+y=8.

Also, when the digits are reversed, the number is increased by 18.

So, the equation is 10y+x =10x+y+18

⇒ 10(8-x)+x =10x+(8-x)+18 (by substituting the value of y) ⇒ 80-10x+x =10x+8-x+18 ⇒ 80-9x=9x+26 ⇒ 18x = 54 ⇒ x=3

Substituting x=3 in y=8-x, we get, ⇒ y = 8-3 = 5 ⇒ 10x+y=10(3)+5 =35 Answer: The required number is 35.

Example 2: Jake's piggy bank has 11 coins (only quarters or dimes) that have a total value of $1.85. How many dimes and quarters does the piggy bank has?

Solution: Let us assume that the number of dimes be x and the number of quarters be y in the piggy bank. Let us form linear equations in two variables based on the given information.

Since there are 11 coins in total, x+y=11 ⇒ y=11-x. We know that, 1 dime = 10 cents and 1 quarter = 25 cents. The total value of the money in the piggy bank is $1.85 (185 cents).

Thus we get the equation 10x + 25y = 185 ⇒ 10x + 25(11-x) = 185 (as y = 11-x) ⇒ 10x + 275 - 25x =185 ⇒ -15x +275 =185 ⇒ -15x=-90 ⇒ x = 6

Substitute this value of x in x+y=11. ⇒ y=11-6=5

Answer: Therefore, the number of dimes is 6 and the number of quarters is 5.

Example 3: In a river, a boat can travel 30 miles upstream in 2 hours. The same boat can travel 51 miles downstream in 3 hours. Find,

  • What is the speed of the boat in still water?
  • What is the speed of the current?

Solution: Let us assume that:

  • the speed of the boat in still water = x miles per hour
  • the speed of current = y miles per hour.

During upstream, the current pulls back the boat's speed and the speed of the boat upstream = (x-y). During downstream, the current's speed adds to the boat's speed and the speed of the boat downstream = (x+y).

Using the last two columns of the table, we can form a pair of linear equations in two variables: x-y=15 x+y=17

Adding both equations we get: 2x = 32 ⇒ x=16

Substitute x=16 in x+y=17 16+y= 17 y=1

Answer: Therefore, the speed of the boat is 16 miles per hour and the speed of the current is 1 mile per hour.

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Practice Questions on Linear Equations in Two Variables

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FAQs on Linear Equations in Two Variables

What is meant by linear equation in two variables.

A linear equation is an equation with degree 1. A linear equation in two variables is a type of linear equation in which there are 2 variables present. For example, 2x - y = 45, x+y =35, a-b = 45 etc.

How do you Identify Linear Equations in Two Variables?

We can identify a linear equation in two variables if it can be expressed in the form ax+by+ c = 0, consisting of two variables x and y and the highest degree of the given equation is 1.

Can You Solve a Pair of Linear Equations in Two Variables?

Yes, we can solve pair of linear equations in two variables using different methods and ensure there are two equations present in the given system of equations so as to obtain the values of variables. If there is one solution it means that the given lines are intersecting, if there is no solution possible, then it means that the given equations are of parallel lines. If there are infinitely many solutions possible, it means that the given equations are forming coincidental lines.

How to Graphically Represent a Pair of Linear Equations in Two Variables?

We can represent linear equations in two variables graphically using the steps given below:

  • Step 1: A system of two equations in two variables can be solved graphically by graphing each equation by converting it to the form y=mx+b by solving the equation for y.
  • Step 2: The points where both lines meet are identified.
  • Step 3: The point of intersection is the solution of the given pair of linear equations in two variables.

How Does One Solve the System of Linear Equations in Two Variables?

We have different methods to solve the system of linear equations:

  • Determinant or Matrix Method

How Many Solutions Does a Linear Equation with Two Variables Have?

Suppose we have a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0. The solutions of a linear equation with two variables are:

  • One and unique if a 1 /a 2 ≠ b 1 /b 2
  • None if a 1 /a 2 = b 1 /b 2 ≠ c 1 /c 2
  • Infinitely many if a 1 /a 2 = b 1 /b 2 = c 1 /c 2

How is a Linear Inequality in Two Variables like a Linear Equation in Two Variables?

A linear inequality in two variables and a linear equation in two variables have the following things in common:

  • The degree of a linear equation and linear inequality is always 1.
  • Both of them can be solved graphically.
  • The way to solve a linear inequality is the same as linear equations except that it is separated by an inequality symbol. But note that the inequality rules should be taken care of.

solve real world and mathematical problems leading to two linear equations in two variables

Linear Equations in the real world

A page on how to find the equation and how to graph real world applications of linear equations.

  • Interpreting Graphs of Real World Linear Equations
  • Worksheet on real world linear equations (worksheet version of this web page)
  • slope of a line
  • y-intercept
  • interactive linear equation
  • equation given slope and a point
  • equation from graph of a line
  • real world applications

A cab company charges a $3 boarding rate in addition to its meter which is $2 for every mile. What is the equation of the line that represents this cab company's rate?

solve real world and mathematical problems leading to two linear equations in two variables

A cab company charges a $5 boarding rate in addition to its meter which is $3 for every mile. What is the equation of the line that represents this cab company's rate?

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A cab company charges a $3 boarding rate in addition to its meter which is $½ for every mile. What is the equation of the line that represents this cab company's rate?

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A cab company charges a $4 boarding rate in addition to its meter which is $ ¾ for every mile. What is the equation of the line that represents this cab company's rate?

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A cab company does not charge a boarding fee but then has a meter of $4 an hour. What equation represents this cab company's rate?

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Mathematics LibreTexts

2: Solving Linear Equations

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In this chapter, you will explore linear equations, develop a strategy for solving them, and relate them to real-world situations.

  • 2.1: Introduction Law enforcement and the military are using drones rather than send personnel into dangerous situations. Building and piloting a drone requires the ability to program a set of actions, including taking off, turning, and landing. This, in turn, requires the use of linear equations.
  • 2.2: Use a General Strategy to Solve Linear Equations Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!
  • 2.3: Use a Problem Solving Strategy Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.
  • 2.4: Solve a Formula for a Specific Variable Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily. It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.
  • 2.5: Solve Mixture and Uniform Motion Applications
  • 2.6: Solve Linear Inequalities
  • 2.7: Solve Compound Inequalities
  • 2.8: Solve Absolute Value Inequalities
  • 2.9.1: Key Terms
  • 2.9.2: Key Concepts
  • 2.10.1: Review Exercises
  • 2.10.2: Practice Test

COMMENTS

  1. Lesson 4

    8.EE.C.8.C — Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Foundational Standards 8.F.B.4 Criteria for Success

  2. Illustrative Mathematics

    • know how to solve real-world and mathematical problems leading to a system of two linear equations in two variables (8.EE.C.8.c). The part of the story happening in this unit: In this unit, students build on what they know from middle school about linear equations and inequalities and systems of linear equations and expand their ...

  3. Illustrative Mathematics

    c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Tasks

  4. Illustrative Mathematics

    No tasks yet illustrate this standard. 8.EE.C.8.c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Quinoa Pasta 1. Summer Swimming.

  5. 8th Grade Math

    8.EE.C.8.C — Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

  6. Expressions & Equations

    Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. CCSS.Math.Content.6.EE.B.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.

  7. Solve real-world and mathematical problems leading to two linear

    Solve real-world and mathematical problems leading to two linear equations in two variables. Popular Tutorials in Solve real-world and mathematical problems leading to two linear equations in two variables. How Do You Use a System of Linear Equations to Find Coordinates on a Map? Like riddles? A word problem is just like a riddle!

  8. Systems of Equations (Word Problems)

    B. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. C. Solve real-world and mathematical problems leading to two linear equations in ...

  9. Standards::Solve real-world and mathematical problems leading to two

    Standards::Solve real-world and mathematical problems leading to two linear equations in two variables. Standards CCSS.Math.Content.8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. CCSS.Math.Content.8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. Expressions and Equations

  10. 4.1 Solve Systems of Linear Equations with Two Variables

    To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x, y) that make both equations true. These are called the solutions of a system of equations. Solutions of a System of Equations

  11. MAFS.8.EE.3.8

    Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

  12. Systems of Equations (Types of Solutions)

    B. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. C. Solve real-world and mathematical problems leading to two linear equations in ...

  13. 13 Solving Real World Problems With Linear Equations in Two Variables

    In this video lesson, you will learn to appropriately use the different mathematical models in solving real life problems involving linear equations in two v...

  14. 2.5.4: Applications Using Linear Models

    Applying Linear Models. Modeling linear relationships can help solve real-world applications. Consider the example situations below, and note how different problem-solving methods may be used in each. Nadia has $200 in her savings account. She gets a job that pays $7.50 per hour and she deposits all her earnings in her savings account.

  15. Systems of Equations (Elimination)

    C. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Common Core: 8.EE.8 Suggested Learning Targets. I can identify the solution(s) to a ...

  16. Solutions to 2-variable equations (video)

    Lesson 1: Two-variable linear equations intro Two-variable linear equations intro Solutions to 2-variable equations Worked example: solutions to 2-variable equations Solutions to 2-variable equations Completing solutions to 2-variable equations Complete solutions to 2-variable equations Math > Algebra 1 > Linear equations & graphs >

  17. 5.9 Systems of Linear Equations in Two Variables

    The steps to use to solve a system of linear equations by graphing are shown here. Graph the first equation. Graph the second equation on the same rectangular coordinate system. Determine whether the lines intersect, are parallel, or are the same line. Identify the solution to the system. If the lines intersect, identify the point of intersection.

  18. 4.1: Solve Systems of Linear Equations with Two Variables

    An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations. {2x + y x − 2y = 7 = 6 { 2 x + y = 7 x − 2 y = 6. A linear equation in two variables, such as 2x + y = 7 2 x + y = 7, has an infinite number of solutions.

  19. Algebra

    For problems 1 - 3 use the Method of Substitution to find the solution to the given system or to determine if the system is inconsistent or dependent. x−7y = −11 5x+2y = −18 x − 7 y = − 11 5 x + 2 y = − 18 Solution 7x−8y = −12 −4x+2y = 3 7 x − 8 y = − 12 − 4 x + 2 y = 3 Solution 3x+9y = −6 −4x −12y = 8 3 x + 9 y = − 6 − 4 x − 12 y = 8 Solution

  20. Linear Equations in Two Variables

    Step 1: To solve a system of two equations in two variables graphically, we graph each equation. To know how, click here or follow steps 2 and 3 below. Step 2: To graph an equation manually, first convert it to the form y = mx+b by solving the equation for y.

  21. Linear Equations in the real world

    equation from graph of a line. real world applications. images. Problem 1. A cab company charges a $3 boarding rate in addition to its meter which is $2 for every mile. What is the equation of the line that represents this cab company's rate? Problem 2. A cab company charges a $5 boarding rate in addition to its meter which is $3 for every mile.

  22. 2: Solving Linear Equations

    2.2: Use a General Strategy to Solve Linear Equations Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to ...