# Lesson 17

Systems of Linear Equations and Their Solutions

### Lesson Narrative

This lesson serves two main goals. The first goal is to revisit the idea (first learned in middle school) that not all systems of linear equations have a single solution. Some systems have no solutions and others have infinitely many solutions. The second goal is to investigate different ways to determine the number of solutions to a system of linear equations.

Earlier in the unit, students learned that the solution to a system of equations is a pair of values that meet both constraints in a situation, and that this condition is represented by a point of intersection of two graphs. Here, students make sense of a system with no solutions in a similar fashion. They interpret it to mean that there is no pair of values that meet both constraints in a situation, and that there is no point at which the graph of each each equation would intersect.

Next, students use what they learned about the structure of equations and about equivalent equations to reason about the number of solutions. For instance, students recognize that equivalent equations have the same solution set. This means that if the two equations in a system are equivalent, we can tell—without graphing—that the system has infinitely many solutions. These exercises are opportunities to look for and make use of structure (MP7).

Likewise, students are aware that the graphs of linear equations with the same slope but different vertical intercepts are parallel lines. If the equations in a system can be rearranged into slope-intercept form (where the slope and vertical intercept become "visible"), it is possible to determine how many solutions a system has without graphing.

### Learning Goals

Teacher Facing

• Determine whether a system of equations will have 0, 1, or infinitely many solutions by analyzing their structure or by graphing.
• Recognize that a system of linear equations can have 0, 1, or infinitely many solutions.
• Use the structure of the equations in a linear system to make sense of the properties of their graphs.

### Student Facing

• Let's find out how many solutions a system of equations could have.

### 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.)

### Student Facing

• I can tell how many solutions a system has by graphing the equations or by analyzing the parts of the equations and considering how they affect the features of the graphs.
• I know the possibilities for the number of solutions a system of equations could have.

Building On