Lesson Plan: Solving Systems of Equations by Graphing

Lesson Summary

This 50-minute lesson focuses on solving systems of equations graphically by identifying intersection points on a graph. Students will learn the limitations of graphical solutions, such as precision. The lesson includes interactive activities, worked examples, and multimedia resources from Media4Math.com to reinforce concepts. Students will conclude the session with a 10-question quiz with an answer key provided.

Lesson Objectives

Solve systems of linear equations graphically.
Interpret solutions as points of intersection on a graph.
Identify limitations of graphical solutions in terms of precision.

Common Core Standards

CCSS.MATH.CONTENT.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs).
CCSS.MATH.CONTENT.HSA.REI.D.10 Understand that the graph of an equation is the set of all solutions.

Prerequisite Skills

Graphing linear equations using slope-intercept form.
Finding the slope and y-intercept of lines.

Key Vocabulary

Graphical Solution: The intersection of two or more lines on a graph.
- Multimedia Resource: https://www.media4math.com/library/22050/asset-preview
Slope: The steepness of a line, represented by m in y = mx + b.
- Multimedia Resource: https://www.media4math.com/library/22184/asset-preview
y-Intercept: The point where a line crosses the y-axis, represented by b in y = mx + b.
- Multimedia Resource: https://www.media4math.com/library/74608/asset-preview

Additional Multimedia Resources

Definition of Linear Systems https://www.media4math.com/library/22088/asset-preview
Tutorial: What Is a System of Equations? https://www.media4math.com/library/36055/asset-preview
Video Definition 20: Linear Function Concepts – System of Linear Equations https://www.media4math.com/library/74618/asset-preview
This collection of math definitions covers the topic of Systems of Equations: https://www.media4math.com/Definitions--LinearSystems
Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

Warm Up Activities (10 minutes)

Choose from one or more of these activities.

1. Desmos Activity

In this activity students will be analyzing the system made up of these two equations:

y = 2x + 1

y = -x + 4

Before accessing Desmos, have students look at the equations, in particular the slopes of the lines, and make a determination if the lines will be intersecting at all. Next, have students launch the Desmos Graphing Calculator.

Instructions:

Enter the equations y = 2x + 1 and y = -x + 4 into the equation editor.
Observe where the lines intersect on the graph.
Zoom in and click on the intersection point to see the solution.
Have students create a label to mark the intersection point. If possible, have them create a screen capture of their system.
Ask students what the intersection means.

Extension: Ask students to adjust one equation (e.g., change y = -x + 4 to y = -x + 3) and predict how the intersection will change before graphing.

2. Image Analysis

Show graphs of intersecting, parallel, and overlapping lines. Discuss how these configurations relate to solutions (one solution, no solution, infinite solutions).

Intersecting Lines

Parallel Lines

Infinite Solutions Systems of Equations

Hands-On Activity

Students use graph paper to manually graph y = x + 2 and y = 2x - 1 and find their intersection.

Multimedia Resources

This collection of math definitions covers the topic of Systems of Equations: https://www.media4math.com/Definitions--LinearSystems
Video Definition: Systems of Linear Equations: https://www.media4math.com/library/74618/asset-preview
Slide Show of terms related to Systems: https://www.media4math.com/library/slideshow/definitions-systems-equations

Teach (25 minutes)

Summary

Students will learn that solving systems graphically involves finding the intersection of lines on a coordinate plane. The precision of the solution depends on accurate graphing.

Instructional Examples

Example 1

Find the solution to this system of equations using graphing:

y = 2x + 1

y = -3x + 4

Graph both equations on a coordinate plane. You can use a graphing calculator. Here are screen showing the TI-Nspire and the Desmos graphing calculator.

TI-Nspire	Desmos

Solution: The lines intersect at (0.6, 2.2).

Verification: Substitute x = 0.6 and y = 2.2 into both equations to confirm.

Example 2

Find the solution to this system of equations:

y = -3x + 5

y = 2x - 1

TI-Nspire	Desmos

Graph the lines and find the intersection at (1.2, 1.4).

Real-World Example

Two delivery service companies have different plans:

Plan A charges 10 cents an item with a \$10 monthly fee.

Plan B charges 20 cents per item with a monthly fee of \$5.

Interpret the solution to this system.

Solution: Write the system of equations. Note that the rate being charged per delivery is the slope of each line.

y = 0.1x + 10

y = 0.2x + 5.

All values to the right of the y-intercept for each graph are the only legitimate values. At x = 50 items, the delivery cost is the same. After that Plan A is less expensive.