# Lesson plan: Solving Graphic Equations with the Intersection Method

This article is of a lesson plan that shows students how to solve systems of equations using the substitution method and the intersection method on the graphing calculator.

## Lesson Overview

In this lesson, students will be introduced to the substitution method to solve systems of equations algebraically, as well as using the intersection method on the graphing calculator.

## Student Objectives

In this lesson, students will:

- Review solving quadratic formulas algebraically
- Learn the substitution method to solve a system of equations
- Learn the intersection method with the graphing calculator to solve a system of equations
- Listen to the teacher give a lecture on these two concepts, and asking questions when appropriate
- Execute sample problems supplied by the teacher and share their answers orally with the rest of the class
- Learn the following new vocabulary words: system of equations, substitution method, intersection method

## Learning Activities

1) Begin the lesson by giving students a brief overview of what they will be learning (solving systems of equations using the substitution method and the intersection method), and explain that the substitution method is an algebraic method and the intersection method is a graphing method.

2) Next, explain the substitution method using a simple pair of equations: x^{2} + y = 0 and 5x = y + 14. Use this process:

- Take the first equation and subtract x from both sides. This will give the following equation: y = -x.
- Take the second equation and subtract 14 from both sides. This will give the following equation: y = 5x - 14.
- Point out to the students that y is equal to both -xand 5x - 14, which gives the equation -x = 5x - 14. This is substitution.
- Add x to both sides, resulting in x + 5x - 14, which can be solved algebraically: (x+7)*(x-2), with x = -7 and 2. When x = -7, y = -49, and when x = 2, y = -4. The answer can be expressed as ordered pairs: (-7, -49), (2, -4).

3) Show how these equations can be solved using the intersection finder on the graphing calculator, using this process:

- Turn on the calculator and press the [Y=] key
- Enter the first equation (y = -x) on the first line
- Enter the second equation (y = 5x - 14) on the second line
- Press the [GRAPH] button to see the graphs (a parabola and a line) on the screen. Adjust the viewing window as necessary so the points at which the two graphs intersect can be seen easily.
- To find the points of intersection: Press [2nd] [TRACE] then 5 to choose "intersect"
- Press the arrow key to move the cursor close to one intersection point and press [ENTER] to select the parabola
- Press the arrow key along the line to move the cursor close to the same intersection point and press [ENTER] to select the line
- Press [ENTER] a third time to have the calculator guess the intersection. Repeat this process with the second intersection point between the parabola and the line.

4) Provide students with increasingly difficult examples, which the students will be required to solve as the lecture progresses, and provide their answers orally.