Introduction to a System of Linear Equations

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Objective

SWBAT state the solutions from a system of linear equations.

Big Idea

Students need to be able to meaningfully describe the solution to a system of linear equations.

Warm up

10 minutes

I plan for students to work on this Warm up for five minutes. Then, we will use an additional five minutes to review as a class. 

A common practice in my classroom is to question students about their prior knowledge as we review the Warm Up. Today, there is a graph of a linear function and I plan to ask the students the following questions:

  • How many solutions does this line have?
  • What does it mean to call a point a solution? 
  • Can you identify 3 solutions on this line?
  • What is the equation of this line?
  • How can you use the equation to check the solutions that you have chosen?

After going over the warm up, I state that the objective for today is to be able to find the solution or solutions to a system of linear equations, which is 2 or more equations describing the same two variables.  I also state that the meaning of a solution will be the same for other functions and graphs that we study this year.  

I model going over the warm up in the video below:

 

 

 

Introduction to Systems

30 minutes

After going over the warm up, I begin the lesson with this PowerPoint, Introduction to a System of Equations.

At the beginning of the PowerPoint, I repeat the concept that each line on a graph represents infinitely many solutions to the equation that the graph represents.  This idea leads into the activity I discuss in the video below to find the common solutions when two lines share a graph to form a system of equations.

At the end of the activity, I provide students with a hard copy of the next page, as notes on each type of solution and the vocabulary associated with each type to place in their notebooks.

In the last part of the PowerPoint, I provide examples of different types of solutions to a system of linear equations.  

Example 1 - Intersecting lines (state the (x,y) intersection point)

Example 2 - A vertical and horizontal line intersecting (Also state the (x,y) intersection point)

Example 2 provides an opportunity to discuss the question, "Will a vertical line and a horizontal line on the same graph always intersect?" I find that discussing this question with respect to Example 2 helps students to comprehend the meaning of Examples 3 and 4 more easily.

Example 3 -  Parallel lines (Must state no solution)

Example 4 -  Same line (Must state infinitely many solutions)

Closure

10 minutes

In this Think Pair Share closure activity, I ask students to work backwards by creating a solution or ordered pair of their choice, and using the Graphs provided to demonstrate lines to the solution. This activity helps students have a clearer understanding of what a solution means and the relationship between the solution, equations of the system and the graph.  

Students are assigned in homogeneous pairs in my classroom and that is how I pair them for this activity, with their assigned partner.

As an extension of this activity, I have students work together with their partner to write 2 equations that have no solution, and 2 equations that have infinitely many solutions.  This allows students individual think time, and the ability to discuss their thoughts with another student.  This process time is important for student learning. Not only does it allow students think time, it provides examples for us to share as a class, the equations they created, and why the reason for their choices.