# MA.8.AR.4.3

Given a mathematical or real-world context, solve systems of two linear equations by graphing.

### Clarifications

Clarification 1: Instruction includes approximating non-integer solutions.

Clarification 2: Within this benchmark, it is the expectation to represent systems of linear equations in slope-intercept form only.

Clarification 3: Instruction includes recognizing that parallel lines have the same slope.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Linear Equation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students determined constants of proportionality and graphed proportional relationships from a table, equation or a written description given in a mathematical or real-world context. In grade 8, students extend this learning to a system of two linear equations and graphing the system on the same coordinate plane then students may determine whether there is one solution, no solution or infinitely many solutions. In Algebra 1, students will write and solve a system of two-variable linear equations algebraically and graphically given a mathematical or real-world context.
• Systems of linear equations can have one solution, infinitely many solutions or no solutions.
• A system of linear equations whose graphs meet at one point (intersecting lines) has only one solution, the ordered pair representing the point of intersection.
• A system of linear equations whose graphs are coincident (the same line) has infinitely many solutions, the set of ordered pairs representing all the points on the line.
• A system of linear equations whose graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. The technical name for these kinds of systems is "inconsistent.”

• A system of linear equations is two linear equations that should be solved at the same time. Instruction includes understanding that systems are on the same coordinate plane to determine solutions (MTR.4.1).
• The purpose of this benchmark is to focus on graphing to solve the system of equations. This allows for the visual representation of what the solution means in context (MTR.7.1).
• Instruction includes recognizing when the system does not have a solution: if there are two distinct lines, but the slopes of the two lines are the same, then the result is a pair of parallel lines. This could be modeled on a graph on paper or through an online resource to support students being able to visualize the lines.

### Common Misconceptions or Errors

• Students make errors in plotting points and graphing lines on the coordinate plane, leading to incorrect solutions. To address this misconception, use graph paper, a printed coordinate plane or an online tool for graphing.
• Students incorrectly identify the solution to equations of the same line by stating only the graphed points are the solution set.
• For example, in the system below with the infinitely many solutions, students may incorrectly not identify (7, 9) as a solution because it is not a point graphed on the coordinate plane.

### Strategies to Support Tiered Instruction

• Instruction includes the use of graph paper, a printed coordinate plane, or an online tool for graphing.
• Teacher provides opportunities for students to comprehend the context or situation by engaging in questions.
• What do you know from the problem?
• What is the problem asking you to find?
• Can you create a visual model to help you understand or see patterns in your problem?
• Instruction includes drawing connections between systems of equations represented graphically and with equations. Using a graphic organizer, reinforce the solution to a system of equation as the ordered pair that satisfies both equations simultaneously.
• When there is one solution, the two lines intersect at one point and when using substitution, the coordinates of that one point will result in true statements for both equations.
• When there is no solution, the two lines do not intersect and therefore there are no coordinates that will result in true statements for both equations.
• When there are infinite solutions, the two lines coincide and intersect with an infinite number of points. When using substitution, all the points on the lines will results in true statements for both equations.
• Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose.
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?

Part A. Graph the line $y$ = 2$x$ + 2.5 on a coordinate plane. Draw two other lines with the same slope but different $y$-intercepts.
Part B. Compare the lines graphed in part A. What do you notice about the other two lines when compared to the given line?

### Instructional Items

Instructional Item 1
Solve the system of linear equations by graphing.
$y$ = $x$ + 5 $y$ = 3$x$ − 3

Instructional Item 2
Solve the system of linear equations by graphing.
$y$ = −$\frac{\text{3}}{\text{4}}$$x$ + 5 $y$ = −4$x$ − 2

Instructional Item 3
Solve the system of linear equations by graphing.
$y$ = 2$x$ + 6 $y$ = 2$x$ − 4.2

Instructional Item 4
Solve the system of linear equations by graphing.
$y$ = −0.5$x$ + 2 = 4.5$x$ − 5

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.AR.4.AP.3: Given two sets of coordinates for two lines, plot the lines on a coordinate plane and describe or select the solution to a system of linear equations.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Solving System of Linear Equations by Graphing:

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

Type: Formative Assessment

Identify the Solution:

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

Type: Formative Assessment

## Lesson Plans

Changes are Coming to System of Equations:

Use as a follow up lesson to solving systems of equations graphically. Students will explore graphs of systems to see how manipulating the equations affects the solutions (if at all).

Type: Lesson Plan

Battle on the High Seas: Applying Systems of Linear Equations:

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

A Scheme for Solving Systems:

Students will graph systems of linear equations in slope-intercept form to find the solution to the system. Students will practice with systems that have one solution, no solution, and all solutions. 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.

Type: Lesson Plan

Where does my string cross?:

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

Type: Lesson Plan

Exploring Systems of Equations using Graphing Calculators:

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.

Type: Lesson Plan

My Candles are MELTING!:

In this lesson, students will apply their knowledge to model a real-world linear situation in a variety of ways. They will analyze a situation in which 2 candles burn at different rates. They will create a table of values, determine a linear equation, and graph eachÂ to determine if and when the candles will ever be the same height. They will also determine the domain and range of their functions and determine whether there are constraints on their functions.

Type: Lesson Plan

Human systems of linear equations:

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.

Type: Lesson Plan

Kimi and Jordan:

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.

Fixing the Furnace:

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.

How Many Solutions?:

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.

## MFAS Formative Assessments

Identify the Solution:

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

Solving System of Linear Equations by Graphing:

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

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Kimi and Jordan:

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.

Fixing the Furnace:

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.

How Many Solutions?:

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.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Kimi and Jordan:

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.