8th Grade Mathematics
UNIT 3: LINEAR EQUATIONS AND SYSTEMS
Unit Description/ Topic Length:
In this 8-week unit, students will analyze, translate, and solve one-variable linear equations
and pairs of simultaneous linear equations, both algebraically and graphically. They will
understand the meaning of the solution to a system and their graphs. Students will build
on what they know about two-variable linear equations and expand the varieties of realworld and mathematical problems they can solve. The mathematical tasks will provide
students an opportunity to connect the content addressed in this unit and the Mathematical
Practices.
Essential Question:
 How do we express a relationship mathematically?
 How do we determine the value of an unknown quantity?
Key Ideas
 When an ordered pair in a system
makes all the equations true, it is a
solution to the system of linear
equations and the point of intersection
on a graph.
 A solution to a system can be
interpreted in the context of a real
world situation involving two
variables.
 Drawing graphs of both equations can
quickly show whether a system of
linear equations has exactly one
solution, no solution, or infinitely
many solutions.
o When two lines are parallel
(same slope and different yintercept), there are no points
of intersection, therefore, there
is no solution for the system.
o When two lines are the same
(overlap each other on a
graph), there are infinitely
many solutions.
 Simple case systems can be solved by
inspection. For example: A system
Guiding Questions:




What is a system of equations?
How can mathematical models (tables,
graphs, equations) be used to display and
describe a real world situation?
What does it mean if an x-value gives the
same y-value for simultaneous equations?
What are the three types of solutions for
systems of equations?
8th Grade Mathematics
where both equations contain the
same slope and different y-intercept
will never intersect, or two equations
written in standard form where both
have the same A and B coefficients
and different constants will not share
a solution.
 When two expressions are equivalent
to the same variable, substitution can
be used to solve the system
algebraically.
 When two equations can be added or
subtracted in such a way that one of
the variables is eliminated,
elimination can be used to solve the
system algebraically.
NYS Common Core Standards for Mathematics Assessed:
Mathematical Content
8.EE.7 Solve linear equations in one variable.
8.EE.7.a Give examples of linear equations in one variable with one solution,
infinitely many solutions, or no solutions. Show which of these possibilities is the
case by successively transforming the given equation into simpler forms, until an
equivalent equation of the form x = a, a = a, or a = b results (where a and b are
different numbers).
8.EE.7.b Solve linear equations with rational number coefficients, including
equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
8.EE.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.8a. Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously.
8.EE.8b. Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by inspection. For
example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot
simultaneously be 5 and 6.
8.EE.8c. Solve real-world and mathematical problems leading to two linear
equations in two variables. For example, given coordinates for two pairs of points,
determine whether the line through the first pair of points intersects the line through the
second pair.
8th Grade Mathematics
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Content
 Linear Equations in One Variable
 Pairs of simultaneous equations
represented as tables, equations, and
graphs
 Pairs of simultaneous equations solved
algebraically: substitution and
elimination
 Pairs of simultaneous equations solved
graphically
 Pairs of simultaneous equations solved
by inspection
 Identifying the type of solution for a pair
of simultaneous equations
 Estimating solutions for a pair of
simultaneous equations on a graph
 Applications of linear systems
 Interpreting the solution for a pair of
simultaneous equations in the context of
the problem.
Skills
 Solve simultaneous equations
graphically and algebraically
 Graph pairs of simultaneous equations.
 Estimate solutions using a graph
 Solve simple cases of pairs of
simultaneous equations by inspection.
 Write simultaneous equations to
represent situations
 Interpret solutions to pairs of
simultaneous equations in the context
of the problem.
 Determine if and when two or more
equations in context have a solution
algebraically and graphically
Vocabulary/ Key Terms
 Equation
 System of linear equations
 Solution of a one-variable equation or system of linear equations
 One solution
 no solution
 Infinitely many solutions
 parallel lines
8th Grade Mathematics
For ELLs, have them create a mini-glossary; instruct them to list each term, its
meaning, and an illustration for each. When students have finished their
glossaries, have them share them with the class. Allow them to use the glossaries
as they work the exercises in the lesson and remind them to use them when they
do their homework.
ASSESSMENT EVIDENCE
Diagnostic and Pre/Post Assessment
 Unit Readiness Test
Formative Assessments:
 Quizzes
 Exit Slips
 Checks for Understanding
 Short- and Extended-Response questions used throughout the unit.
 Reflections
 Formative Assessments 1 & 2, FAL (MARS)
Summative Assessments:
1. FINAL PERFORMANCE TASK: TALK AND TEXT PLANS (NYC Common Core
Library)
The task asks students to:
o Write an equation for Talk and Text Plan B using the variables provided.
o Solve algebraically for the number of minutes equally shared for both plans.
o Graph each Talk and Text plan and determine the shared cost of the plans for
the number of minutes shared.
o Explain how the graph matches the algebraic work/solution.
o Justify which plan would be best to use when only spending $75 using a
method of choice.
2. Unit test
TEACHING PLAN
Teaching and Learning Activities:
I.
Administer the Unit Readiness Test.
8th Grade Mathematics
II.
III.
Instruction follows the Launch-Explore-Summarize flow.
Use the guiding questions to focus each lesson.
 Lesson 1: Solving Linear Equations in One Variable (15 days)
 Lesson 2: Solving Simultaneous Equations or Systems of Linear Equations by
Graphing [5 days]
 Administer Formative Assessment #1.
1. Consider the equation 5x−2y=3.
a. If possible, find a second linear equation to create a system of equations
that has indicated number of solutions. Use mathematical reasoning to
support your answer.
i.
Exactly 1 solution.
ii.
No solution.
iii.
Infinitely many solutions.
b. In each case, how many such equations can you find?
2. Kimi and Jordan are each working during the summer to earn money in
addition to their weekly allowance, and they are saving all their money. Kimi
earns $9 an hour at her job, and her allowance is $8 per week. Jordan
earns $7.50 an hour, and his allowance is $16 per week.
a. Complete the two tables shown below.
b. Write an equation that can be used to calculate the total of Kimi's allowance
and job earnings at the end of one week given the number of hours she
works.
c. Write an equation that can be used to calculate the total of Jordan's
allowance and job earnings at the end of one week given the number of
hours worked.
d. Sketch the graphs of your two equations on one pair of axes.
e. Jordan wonders who will save more money in a week if they both work
the same number of hours. Write an answer for him.

Lesson 3: Solving Simultaneous Equations or Systems of Linear Equations by
Substitution [8 Days]
 Administer Formative Assessment #2.
1. Solve each system by substitution. Check your solution.
a. y = x - 2
2x + 2y = 4
8th Grade Mathematics
b. 0.5x – y = -4
-x + 2y = 8
2. Two brothers decide to save money. One starts with $10, and saves $2 each day.
The other starts with none, but saves $3 each day.
Part A Write two equations to represent both brothers’ savings plans.
Part B Will they ever have the same amount of money? If not, explain why. If they
will, after how many days will they have the same amount?
IV.
V.
VI.
VII.
Administer the MARS Formative Assessment Lesson (FAL) titled Classifying Solutions to
Systems of Equations. [2 Days]

Lesson 4: Solving Simultaneous Equations or Systems of Linear Equations by
Elimination [5 Days]

Lesson 5: Applications of Linear Systems [5 Days]
Assess students on the unit by administering the Final Performance Assessment and the
Unit test.
Use the essential question as a post-assessment.
Administer the unit test.
Resources and Materials Needed:
 Connected Math Project 3 (CMP3)
Unit: It’s in the System; Say It With Symbols
 NYS Common Core Math Module 1: Integer Exponents and Scientific Notation
 Impact Mathematics Course 3
 Integrated Algebra Textbook
 MathXL (Pearson’s online homework, tutorial, and assessment system)
 Scientific Calculators
 Graph Papers
8th Grade Mathematics
CALENDAR
Time Spent
on Standard
Standards
Topics To Cover
Resources
3 weeks
8.EE.7

Solving Linear Equations
in One Variable

Connected Math Project 3
(CMP3)
Unit – Say It With
Symbols: Making Sense of
Symbols
[8.EE.7, 8.F.3, 8.F.4, 8.F.5,
8.G.9]
5 weeks
8.EE.8

Solving pairs of
simultaneous equations
by graphing
o Graphing the
equations in slopeintercept form and
estimate solutions
on a graph for real
life situations
o Investigating
different types of
solutions by using
a graphing
calculator
Solving systems of
equations algebraically by
using substitution
Solving systems of
equations algebraically by
using elimination
Writing and solving
systems of equations in a
real world context using
substitution and
elimination

Connected Math Project 3
(CMP3) Unit – It’s in the
System: Systems of Linear
Equations and
Inequalities
[8.EE.8]



8th Grade Mathematics
FINAL PERFORMANCE TASK (NYC Common Core Library)
Talk and Text Plans
A cell phone company offers two talk and text plans. The company charges a monthly service fee of
$20 for either plan the customer chooses:
Customers that choose Talk and Text Plan A are charged five cents a minute and twenty
dollars for 250 texts.
Customers that choose Talk and Text Plan B are charged ten cents a minute (first 100 minutes
free) and fifteen dollars for 200 texts. The equation: c = .10(m – 100) + 15 + 20 can be used to
represent how much a customer would spend monthly for the minutes used.
a) Express Plan A as an equation where c equals the cost and m equals the minutes used.
b) For how many minutes will both plans share the same cost? Show your work algebraically.
c) Graph each Talk and Text plan to determine when both plans cost the same. Write the solution
and explain how the graph results match your algebraic solution.
d) If a customer has $75 to spend each month, which plan should the customer choose and why?
Use your work to justify your answer.