CURRIKI ALGEBRA UNIT 2
Linear and Exponential Relationships
Lesson 2.1: Solving Systems of Linear Equations Assessment
Unit 2: Linear and Exponential Relationships
This unit may be used for a review of eighth grade algebra and extension to ninth grade
algebra I. In earlier grades, students define, evaluate, and compare functions, and use
them to model relationships between quantities. In this unit, students will learn function
notation and develop the concepts of domain and range. They move beyond viewing
functions as processes that take inputs and yield outputs and start viewing functions as
objects in their own right. They explore many examples of functions, including
sequences; they interpret functions given graphically, numerically, symbolically, and
verbally, translate between representations, and understand the limitations of various
representations. They work with functions given by graphs and tables, keeping in mind
that, depending upon the context; these representations are likely to be approximate
and incomplete. Their work includes functions that can be described or approximated by
formulas as well as those that cannot. When functions describe relationships between
quantities arising from a context, students reason with the units in which those
quantities are measured. Students explore systems of equations and inequalities, and
they find and interpret their solutions. Students build on and informally extend their
understanding of integer exponents to consider exponential functions. They compare
and contrast linear and exponential functions, distinguishing between additive and
multiplicative change. They interpret arithmetic sequences as linear functions and
geometric sequences as exponential functions.
The nine lessons in unit 2 (2.1-2.9) provide the instruction and practice that supports the
culminating activity in the final two-day unit project.
Lesson 2.1: Solving Systems of Linear Equations using Substitution
In this lesson we build upon student experience with graphing and solving systems of
linear equations from middle school to focus on justification of the methods used. We
rove that given a system of two equations in two variables, replacing one equation by
the sum of that equation and a multiple of the other produces a system with the same
solutions.
Common Core State Standards by Cluster
Grade Level Cluster
9
Solve systems of equations
9
Represent and solve
equations and inequalities
graphically
CCS Standard
A.REI.6
A.REI.10
Lesson Preparation Resources for Teachers
Systems of Linear Equations (Winpossible) (video mini-lesson)
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Solving Linear Systems by Substitution (Sal Kahn, video lesson)
Special Types of Linear Systems (Sal Kahn, video)
Solving Systems with Substitution Method (Jesse Mercer; lesson)
Linear Systems (Sal Kahn, video)
TE_Vocabulary Page
TE Assessment
TE Sets of Linear Equations
TE Solving Linear Equations by Substitution Assessment Answer Key
Systems of Equations Baseball (Alice Keeler in Curriki)
Instructional Materials for Students (Print one copy for each student)
SE_Vocabulary (blank)
SE Assessment
SE Sets of Systems of Linear Equations
SE Solving Linear Equations by Substitution Assessment
Time: 50-minute session
Lesson Objectives:
Students will be able to:
 Create equations to represent a given situation
 Solve equations and explain process and reasoning
 Create and solve systems of linear equations
 Graph linear equations to check solutions
Lesson Content
1. Background Building Activity for Students (5 minutes)
a. Vocabulary Building:
Print out one copy of the Vocabulary Page for each student. Review the linear
systems vocabulary words prior to the warm-up problem. Students create their
own definitions including a visual representation.
b. Warm-Up problem:
Ask students to solve the following equations in terms of a different variable.
• Given: A = lw, find the value of l in terms of A & w.
• Given: speed = distance/time, find the value of d in terms of s and t.
• Given: C = 5/9(F - 32), find the value of C in terms of F.
Discuss that equations can be rearranged to solve for the value of any variable in
the equation in terms of any other variable. Students’ solutions should be:
• l = A ÷w
• distance = speed × time
• F = 9/5(C) + 32
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Being able to rearrange and manipulate equations is an important skill. This skill
will be used to solve systems of linear equations using substitution.
2. Focus Question based on today’s lesson (25 minutes)
Today’s focus question (write it on the board)
How can I solve a system of linear equations using substitution?
There are many ways to solve systems of linear equations. Today the focus is on
using substitution. Students will rewrite one of the equations in terms of one of the
variables then substitute the value of that variable into the second equation and
then solve. This method works because after substituting there is only one variable
in the equation and its value can be found. This value then can used to find the value
of the second variable.
a. Whole Class Activity:
Present class with the two linear equations:
6y = 4x - 3 and 7y = 5x + 2
Ask half of the class to rearrange one equation in terms of x and the other half to
rearrange the other equation in terms of x. Next, have students substitute the
value for x into the other equation and solve for y. Once the value of y is
determined, substitute this value back into any of the equations to find the value
of x.
Ask students on each half of the class share the values of x and y they found.
Compare the values for x and y. Explain that either way, the same values for x
and y will be found.
b. Small Group Practice Activity: (Teacher checks for understanding during this
activity)
1. Provide students groups with one system of linear equations from the
worksheet: Sets of Systems of Linear Equations. Ask students to solve for the
values of the variables using substitution. While students are working the
teacher checks for understanding by observing students while they are
working in small groups.
2. Prompt student groups to think about: How did you determine what variable
to solve for? Are there any helpful hints to make the problem easier to solve?
Discuss as a class how students decided what variable to solve for.
Encourage students to check answers by graphing both equations and finding
the point of intersection. Solutions can be checked using the Sets of Systems
of Linear Equations Answer Key.
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3. Present the activity, Systems of Equations Baseball, through projecting the
PowerPoint presentation created by Alice Keeler, found in Curriki. Diagrams
and directions are on the PowerPoint.
In this activity the class is divided into two teams. They simulate a baseball
game by having the batter come up to the board. Arrange 3 chairs like a
baseball diamond in the room where the board is home plate. Each chair
represents a base. Let the batter pick their pitch (1st, 2nd, 3rd base or home
run).
Using your classroom textbook, select questions that are easy, medium, hard
and difficult. When the “batter up” student gets the question correct, he/she
moves to the appropriate base. If students are already on base they must be
FORCED from their base, NO STEALING.
If a student is on 2nd and a student chooses 1st base then the student on 2nd
remains on 2nd. If the student at the board chooses 2nd base then if the
student gets the answer correct the 2nd base student will be moved to 3rd
base. If the student at the board chose 3rd base then the 2nd base student
would be forced home.
If the student gets the question WRONG then it is a pop fly and the ball
needs to be caught by the other team. The other team works together to
come up with an answer. A pre- determined team captain can speak for the
group about the answer. If the other team gets it correct then everyone
from the first team takes their seat and the other team is up to bat. (There
should be a pre determined batting order.)
Keep score. Every time a student makes it back to home plate, the team gets
a point.
c. Individual Activity:
Provide each student with the following word problem:
Hot dogs cost $5.50 and sodas are $3.75. One day the sales at the baseball game
totaled $1,165.00. A total of 257 items were sold. How many hotdogs and how
many sodas were sold?
1) Write 2 equations representing the given information.
2) Solve this system of linear equations using substitution.
3) Graph each equation
4) Identify the point of intersection. How does this point relate to the solution
from step 2?
3. Whole class discussion
a. Students share solutions:
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Select a few students to share their solution to the baseball field concession stand
problem. Students explain how they solved the problem.
**The correct answer is 115 hot dogs and 142 sodas.
b. Algorithm:
To solve a system of linear equations using substitution:
a) Solve one of the equations for the value of one variable in terms of the
other.
b) Substitute this value into the second equation. (This will result in the
equation having a single variable)
c) Solve for the value of the one variable.
d) Substitute the value of the found variable into either of the original
equations to find the value of the second variable.
e) Write the solution as an ordered pair (x, y)
Realize the solution set is the point where the two lines intersect. If the
solution is the empty set, the lines are parallel. If the solution set is all
values of (x, y), then the equations name the same line.
4. Assessment Activity (5 minutes)
Provide each student with a copy of the Solving Linear Equations by Substitution
assessment. Allow time to complete. Evaluate student understanding by correcting
using the provided answer key.
5. Extension Activities:
Solving systems using linear combinations
Using linear combinations is another method to solve systems of linear equations.
This link describes how to solve systems of linear equations using linear
combinations. This method also employs substitution.
Graphing linear equations
After solving systems of linear equations, students should graph the equations to see
the solution set can really help students better understand the purpose for solving
linear equation systems.
Special Types of Linear Systems (Sal Kahn, video)
Ask students to watch this video, then discuss when the solution set will be infinite,
and when the solution set is the empty set. Create 3 sets of linear systems, one for
each possible outcome (one solution, infinite solutions, no solutions)
6. Homework assignment for additional independent practice
Determine if a system of equations has infinite solutions
There are three possible results when solving systems of linear equations. The
solution may be a single point, and infinite set of solutions, or no solution. This
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activity describes when a solution set will have infinite solutions (when the equations
name the same line)
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