Lesson 12

Writing and Graphing Systems of Linear Equations

Lesson Narrative

This is the first of a series of lessons in which students review what they learned about systems of equations in middle school and develop new techniques for solving them.

In this lesson, students recall that a system of equations in two variables is a set of equations that represent multiple constraints in the same situation, and that a solution to the system is any pair of values that satisfy all the constraints simultaneously. Students also revisit the idea that the solution, if there is one, can be represented graphically as the intersection of the graphs of the equations.

Students write a system of equations to represent the quantities and constraints in each of several situations, find a solution that meets multiple constraints by graphing, and then interpret the solution in context. In the process, they reason both abstractly and quantitatively (MP2). As they analyze relationships mathematically and reflect on the results, students also engage in aspects of modeling (MP4).

Some students who recall the work on systems of equations from grade 8 may choose to solve the systems algebraically. This is appropriate and welcome, but it is not necessary to introduce the idea to the class here. Students will use algebra to solve systems starting in the next lesson.

Learning Goals

Teacher Facing

  • Solve systems of linear equations by reasoning with tables and by graphing, and explain (orally and in writing) the solution method.
  • Understand that the solution to a system of equations in two variables is a pair of values that simultaneously make both equations true, and that it is represented by the intersection point of the graphs of the equations.
  • Understand that two (or more) equations that represent the constraints on the same quantities in the same situation form a system.

Student Facing

  • Let’s recall what it means to solve a system of linear equations and how to do it by graphing.

Required Materials

Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Learning Targets

Student Facing

  • I can explain what we mean by “the solution to a system of linear equations” and can explain how the solution is represented graphically.
  • I can explain what we mean when we refer to two equations as a system of equations.
  • I can use tables and graphs to solve systems of equations.

CCSS Standards

Building On


Building Towards

Glossary Entries

  • solution to a system of equations

    A coordinate pair that makes both equations in the system true.

    On the graph shown of the equations in a system, the solution is the point where the graphs intersect.

  • system of equations

    Two or more equations that represent the constraints in the same situation form a system of equations. 

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