Lesson 14

Solving More Systems

Lesson Narrative

In previous lessons, students have worked with contexts where two quantities are changing at different (or possibly the same) rate, and they must find when they are equal. Such systems are represented by equations of the form \(y = mx + b\) and are solved by setting the two expressions for \(y\) equal to each other.

In this lesson, students graduate to other types of systems with different structures. They learn that examining structure is a good first step since it is sometimes possible to recognize an efficient method for solving the system through observation. They see that if at least one of the equations has a single variable isolated, then that expression can be substituted into the other equation in place of \(y\) or \(x\) to get a single equation in one variable that can be solved. Finally, students use the structure of a system of equations to reason about its lack of solutions.

When students look at the structure of a system before starting to solve it in order to develop a good approach to solving, they engage in MP7.

Learning Goals

Teacher Facing

  • Calculate values that are a solution for a system of equations, and explain (orally) the solution method.
  • Generalize (orally) a process for solving systems of equations using substitution.
  • Justify (orally and in writing) that a particular system of equations has no solutions using the structure of the equations.

Student Facing

Let’s solve systems of equations.

Learning Targets

Student Facing

  • I can use the structure of equations to help me figure out how many solutions a system of equations has.

CCSS Standards

Building On

Addressing

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 \(5-(\text-7)=12\).