# Lesson 13

Solving Systems of Equations

### Lesson Narrative

In this lesson, students continue to explore systems where the equations are both of the form $$y = mx + b$$. They connect algebraic and graphical representations of systems, first by matching graphs to systems, then by drawing their own graphs from given systems. Additionally, students see how to see the number of solutions from both the graphical and algebraic representations. In the graphical representation the number of solutions is equal to the number of points where the graphs intersect. In the algebraic representation, two equations with the same rate of change can have 0 or infinitely many solutions depending on whether the initial values ($$y$$-intercepts) are the same or not. If the rates of change are different then there is a single solution, which can be interpreted physically as the point at which two quantities changing at different rates become equal.

### Learning Goals

Teacher Facing

• Coordinate (orally) the solution of an equation with variables on each side to the solution of a system of two linear equations.
• Create a graph of a system of equations, and identify (orally and in writing) the number of solutions of the system of equations.

### Student Facing

Let’s solve systems of equations.

### Required Preparation

Print the Different Types of Systems blackline master. Prepare one set for every 2–3 students. Provide access to straightedges for drawing accurate graphs and scissors for groups that wish to cut apart the graphs on the blackline master.

### Student Facing

• I can graph a system of equations.
• I can solve systems of equations using algebra.

### Glossary Entries

• system of equations

A system of equations is a set of two or more equations. Each equation contains two or more variables. We want to find values for the variables that make all the equations true.

These equations make up a system of equations:

$$\displaystyle \begin{cases} x + y = \text-2\\x - y = 12\end{cases}$$

The solution to this system is $$x=5$$ and $$y=\text-7$$ because when these values are substituted for $$x$$ and $$y$$, each equation is true: $$5+(\text-7)=\text-2$$ and $$5-(\text-7)=12$$.