ALEX | Alabama Learning Exchange

Solving Systems of Linear Equations -- Part 2 Substitution

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This lesson provided by:

Author:	Morgan Boyd
Organization:	Retirement

General Lesson Information

Lesson Plan ID:	35621
Title:	Solving Systems of Linear Equations -- Part 2 Substitution
Overview/Annotation:	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.

Associated Standards and Objectives

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 8

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

a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.

Unpacked Content

Evidence Of Student Attainment:

Students:

Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.

Teacher 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.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
M.8.12.1: Define variables.
M.8.12.2: Recall how to estimate.
M.8.12.3: Recall how to solve linear equations.
M.8.12.4: Demonstrate how to graph solutions to linear equations.
M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
M.8.12.7: Define simultaneous.
M.8.12.8: Recall how to solve linear equations.
M.8.12.9: Recall properties of operations for addition and multiplication.
M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
M.8.12.11: Define point of intersection.
M.8.12.12: Recall how to solve linear equations.
M.8.12.13: Demonstrate how to graph on the Cartesian plane.
M.8.12.14: Identify ordered pairs.
M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
M.8.12.16: Create a word problem from given information.
M.8.12.17: Recall how to solve linear equations.
M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.

Prior Knowledge Skills:

Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
Demonstrate an understanding of an extended coordinate plane.
Draw a four-quadrant coordinate plane.
Draw and extend vertical and horizontal number lines.
Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
Recall how to graph points in all four quadrants of the coordinate plane.

Alabama Alternate Achievement Standards

AAS Standard:
M.AAS.8.12 Solve two-step linear equations where coefficients are less than 10 and answers are integers.

Local/National Standards:

Primary Learning Objective(s):

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.

Preparation Information

Total Duration:	31 to 60 Minutes
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 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 iPad, Chromebook, or MacBook for every student -- to graph the lines to check for accuracy
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.

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

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.

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.

ALEX Lesson Plan

Solving Systems of Linear Equations -- Part 2 Substitution