Solving Systems of Linear Equations -- Part 2 Substitution

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

Overview

This lesson is the second part of solving systems of linear equations. The lesson will be taught in one class period. The concept for the lesson is to solve one equation in terms of "x" or "y" and substitute the results into the other equation. Calculating the final solution to the system will take a few more steps. This lesson will describe the remaining steps as well as examples to follow.

This lesson results from the ALEX Resource Gap Project.

Mathematics (2019) Grade(s): 8

MA19.8.12
Solve systems of two linear equations in two variables by graphing and substitution.

Vocabulary

System of linear equations
Point of intersection
One solution
No solution
Infinitely many solutions
Parallel lines
Slope-intercept form of a linear equation
Standard form of a linear equation

Knowledge

Students know:

The properties of operations and equality and their appropriate application.
Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
Substitution techniques for algebraically finding the solution to a system of linear equations.

Skills

Students are able to:

generate a table from an equation.
Graph linear equations.
Identify the ordered pair for the point of intersection.
Explain the meaning of the point of intersection (or lack of intersection point) in context.
Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.

Understanding

Students understand that:

any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.

Primary Learning Objectives

The student will be able to solve systems of linear equations using the substitution property.

Additional Learning Objective(s)

The student will be able to solve multi-step equations with one variable.

The student will be able to determine if the system of linear equations will have a solution.

Procedures/Activities

Before:

As students enter the room, the teacher will display the bell ringer on the interactive whiteboard. The students will write their answers on their own paper. The teacher will make an informal assessment. Students must be able to solve multi-step equations in order to solve systems of equations by substitution. The teacher will solicit students to answer the bell ringer.

During:

*The lesson will be taught in one class period.*

1. The teacher will start the video from Khan Academy: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-with-substitution/v/the-substitution-method.

2. The teacher can show many types of examples. The website has lines in standard form as well as y-intercept form. The teacher will stop the video and revisit the concept of solving in terms of "y" or "x".

Example:

3x + 6y = -9

3x + 6y - 6y = -9 - 6y subtract 6y from both sides

3x = -6y - 9 put the "y" term in front

3x/3 = -6y/3 - 9/3 divide everything by 3

x = -2y - 3 solved for "x" in terms of "y"

3. The teacher will do an informal assessment to check for understanding. The teacher will start the video again.

4. As the video finishes the last example, the teacher will open the website, DESMOS.com (https://www.desmos.com/calculator). The teacher will type in the two equations and see where the two lines cross. The teacher will show the students that the solution will be the same. The teacher will tell the students that the method does not change the solution.

5. The teacher will hand out the worksheet, "Equations by Substitution" from the attachment section. The teacher will allow the students to work in groups with at least two technology devices. The students must show their work. One of the students in the group will use online graphing tool to check the answers. The teacher will monitor the groups and their discussions. The teacher may extend the time for some groups or assign a peer-tutor.

6. The teacher will check for understanding by asking the groups to work the problems on the interactive whiteboard.

*The lesson concludes day 2 and the next day will be elimination.*

After:

The teacher will display the exit slip from the attachment section on the interactive whiteboard. The students will answer on their own paper and turn in when they leave the room.

Assessment Strategies

Informal:

The teacher will assess the students' understanding with the bell ringer and the video. The teacher will monitor the students in the groups to check for accurate use of terminology as well as comprehension of the mathematical concept.

Formal:

The teacher will use the exit slip for the formal assessment. The student will show all the work to explain the solution.

Acceleration

The accelerated students will have a challenging worksheet to complete. The assignment is in the attachment section called "Challenging Equations".

Intervention

The teacher will place students in groups of 4. The teacher will assign a peer-tutor. The teacher will let the groups use a technological device. The student will use the online graphing website to check their answers. The teacher can extend the time or shorten the number of problems.

Total Duration

31 to 60 Minutes

Background/Preparation

Teacher:

The lesson will be taught in one class period. The teacher will need to preview the video from Khan Academy. The website has many example problems, and the teacher can use as many as necessary. This lesson needs to be taught after introducing the first part of the lesson, "System of Equations -- Part 1 Graphing."

Student:

The student will need to know how to solve multi-step equations. When an equation has two variables, the student will need to know how to solve in terms of one variable. The student will need to be familiar with the Substitution Property. The substitution property of equality states that if x = y, then x can be substituted in for y in any equation, and y can be substituted for x in any equation. The following is an example:

x = 2 and y = 4

2x + 3y -8

evaluate .... 2(2) + 3(4) - 8 = 4 + 12 - 8 = 8

Substitution means to replace the variable in one equation with what the variable is equal to from the other equation.

Materials and Resources

Notebook

Pencil

Calculator

Websites:

https://www.desmos.com/calculator

https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-with-substitution/v/the-substitution-method

Devices:

iPad, Chromebook, or MacBook for every student

Worksheets:

Bell Ringer Substitution -- for the interactive whiteboard -- attachment section

Equations by Substitution -- make copies for all students -- attachment section

Exit Slip Substitution -- for the interactive whiteboard -- attachment section

Accelerated Systems of Equations -- make copies for accelerated students -- attachment section

Technology Resources Needed

Desktop computer with interactive whiteboard for teacher