# MA.8.AR.4.2

Given a system of two linear equations represented graphically on the same coordinate plane, determine whether there is one solution, no solution or infinitely many solutions.
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 represented graphically on the same coordinate plane then students will determine whether there is one solution, no solution or infinitely many solutions. In high school, students will write and solve a system of two-variable linear equations algebraically and graphically given a mathematical or real-world context.
• The focus of this benchmark is on the understanding that the solution of a system is a set of points that satisfy both equations of the system.
• 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).

### Common Misconceptions or Errors

• Students may incorrectly interpret the solution when the lines are the same and have an infinite number of solutions. To address this misconception, provide multiple examples to show how the equations and graphs will be the same line on the coordinate plane.

### Strategies to Support Tiered Instruction

• Instruction includes testing possible solutions for a given system of linear equations to demonstrate whether the equations have the same solution set, one common solution (only one ordered pair) or no common solution.
• 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 stratifies 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.

Ashley looks at the following system of equations. She concludes that because there is no  y - intercept value, the lines cannot intersect.

$y$ = −$\frac{\text{1}}{\text{2}}$$x$ $y$ = $\frac{\text{1}}{\text{3}}$$x$
• Part A. Graph the system of equations on a coordinate plane.
• Part B. Is Ashley's conclusion correct? Explain your answer and support your reasoning with mathematical examples.

Part A. Identify the solution of the graphed system of equations. Explain why it is the solution.
Part B. Identify the solution of the graphed system of equations. Explain how you know it is the solution. What conjecture can you make about parallel lines?

### Instructional Items

Instructional Item 1
Determine the number of solutions of each graphed system of linear equations, A, B and C.

*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.2: Given a system of two linear equations represented graphically on the same coordinate plane, identify whether there is one solution or no solution.

## Related Resources

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

## 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

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

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

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

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.

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.

## 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.

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.