Instructional Leadership Initiative: Supporting Standards-based Practice
Unit:
Algebra 1: System of Equations (Grade 9)
School:
McClymonds High School
Strengths
The teacher-to-teacher notes are very complete and very informative. The annotations on the
examples of student work provide analysis of students’ facility with the standard’s underlying
skills and concepts. The assessment provides enough scaffolding so that students with minimal
levels of competency are still able to complete problems and so that analysis for reteaching
purposes is possible.
Concerns
A major problem is evident in the scoring rubric used to ascertain whether students have or have
not met the standard. The separate levels of performance for students deemed to have met the
standard appear to be related to a grading guide and are not immediately helpful to a scorer. It is
unclear whether students must have 9 of 10 problems of each type correct to warrant a Level 6 or
only 90% of the assessment points. The descriptions provided for the levels below meeting the
standard are much clearer and more informative. The comments on the papers themselves
provide clarity as to why a student has or has not met the standard but seem unaligned with the
rubric.
Samples of student work do not contain examples of the same problems being completed by
students assessed at the various levels of the rubric.
Algebra 1: Systems of Equations
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I.
BACKGROUND
Unit Title:
SYSTEMS OF EQUATIONS
Unit Designers:
McClymonds High School
Discipline/Course Title:
Timeframe:
Grade Level: 9
Algebra 1
13-15 days
Teacher to Teacher Notes:
Students will cover:
Graphical representation of simultaneous equations
Systems of equations that are dependent, consistent and
independent
Substitution and Elimination Methods to solve simultaneous
equations
Choosing the most appropriate method for solving systems of
equations
How to graphically represent a system of equations and
identify a
Solution set/feasible region.
Applying Systems of Equations to application problems
Informative and detailed
description of what is
contained within this unit.
A system can have any number of equations and any number of
variables. This unit will focus primarily on systems that have two
variables and two equations. When there are only two variables it
is possible to visualize the situation with a graph on a coordinate
plane. In this unit we students will use symbolic methods and
tables to find solutions to systems.
The emphasis of this unit is to write and solve systems of
equations. Students learn how to model real-life situations with
two variables using a system of equations or inequalities. They are
introduced to a variety of techniques for solving these systems.
Students will learn how to solve a system of equations graphically,
using substitution and elimination. Inequalities in one variable are
introduced in the unit in lesson six. In lesson seven students will
use two variables. They will also perform operations on
inequalities and learn why multiplying by a negative number on
both sides of the inequality reverses the sign of the inequality and
how to check that the solutions are correct. In subsequent lessons
students will learn that a system of inequalities has an infinite
number of solutions.
McCymonds Algebra 1: Systems of Equations
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Printed Materials Needed:
Algebra 1 Text Book and Algebra 1 Practice Masters (Prentice
Hall)
Algebra Concepts and Applications (Glencoe) and Algebra 1
Reteaching Masters (Holt, Rinehart and Winston).
Resources (non-print):
Graph paper, rulers, pencils, calculators, and overhead projector.
Internet Resources:
Student Tutorial CD-ROM and Resource Pro with Planing Express
CD-ROM and Extra Help on the Web at www.phschool.com and
computer item generator.
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Western Assessment Collaborative at
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II. CONTENT STANDARDS ADDRESSED
The required content knowledge
State/District:
Title:
State of California/ Oakland Unified
School District
Algebra I, Solving Systems of Equations
This unit promotes a better understanding of the methods for
solving systems of linear equations. To this end, the lessons
built on and reinforce student understanding of linear equations
and parallel lines from the following portions of Algebra I
Standards 6, 7 and 8. However the major focus of the unit is on
in-depth study of Standard 9.
Standard 6. Students graph a linear equation and compute the xand y-intercepts.
Standard 7. Students verify that a point lies on a line, given an
equation of the line. Students are able to derive linear equations
by using the point-slope formula.
Standard 8. Students understand the concepts of parallel lines
and how their slopes are related.
Algebra 1:
Standard 9.0
Students solve a system of two linear equations in two variables
algebraically and are able to interpret the answers graphically.
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Unpack the Standard(s)
The knowledge and skills that students need to know and be
able to do to meet the standard.
A. Correctly graph a set of equations on the same Cartesian
coordinate system
1. Set up and label coordinate system
2. Provide appropriate scale
3. Represent given equation graphically, accurately and
neatly
4. Plot points accurately and neatly
The specificity of the
characteristics makes it
clear what is required in
relation to the assessment.
B. Identify the solution set from the graph as an ordered pair
1. Explain clearly parallel, identical and intersecting lines
C. Solve systems of linear equations using substitution.
D. Solve systems of linear equations using elimination/linear
combination
E. Identify the most appropriate method for solving a system
F.
Solve application problems involving a system of linear
equations
1. Accurately identify the variables before setting up the
problem and translating the word problem into an equation.
2. Correctly translate word problems into appropriate
equations.
3. Solve the system accurately using the most appropriate
method and labeling the answer using the correct units.
Enabling Prerequisite Skills:
Basic arithmetic
Solve one and two-step linear equations
Properties of equality
Understanding Cartesian coordinate systems (labeling axes,
plotting points, ordered pairs, scaling axes, etc.
Equations for straight line and forms of linear equations
Understanding of slope
Parallel and perpendicular lines and their equations
Vocabulary: simultaneous, solution set
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III. THE ASSESSMENT
Evidence that students will provide to demonstrate
that they have met the standard.
Type(s) of Evidence Required to Assess the
Standard(s):
Evidence of ability to:
Graph Linear equations and systems of equations in one and
two
variables
Identify solution sets for systems of equations from graphs
Identify equations that are parallel/inconsistent,
perpendicular/consistent, dependent and independent
Interpret algebraic, graphic and verbal representations of
systems of equations
Solve systems of equations using the method of substitution
Solve systems of equations using the method of elimination
Graph solution sets after solving systems of equations
algebraically
Set up and solve systems of two linear equations from word
problems and interpret the solution in the context of the
problem.
The specificity of this list
seems comprehensive and
well-aligned to the previous
definition of the standard.
Assessment Method(s):
Classroom observations / activities
Teacher, student questions
Corrected Class work
Homework
Tests
Teacher to Teacher Notes:
The assessment will consist of ten questions. These questions will
seek to determine the students’ ability to correctly use the various
methods of solving systems of equations. As highlighted in
California Algebra 1 Standard 9.0.
Assessment Prompt(s):
See attached assessment.
McCymonds Algebra 1: Systems of Equations
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Assessment
California Content Standards
Mathematics
SYSTEMS OF EQUATIONS
Algebra I
Standard 9
Students solve a system of two linear equations in two
variables algebraically and are able to interpret their answers
graphically.
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Directions: Read each problem carefully and show all
steps that lead to your solution.
These directions could
appear on either the
coversheet or the first page
to conserve paper.
Write your answers in the space provided.
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1. Determine whether the given point is a solution of the
system of equations. Show your work.
a) (2,8)
y = 3x + 2
b)
y = 9x - 10
(-4,6) y = x – 2
y = -7x + 22
c) (10, -2) 7y + x = 14
d)
x = 10
Providing space for
students to work the
problem that is adjacent to
the problem would enable
teachers to more easily
evaluate the work and
would direct students as to
where to show their work.
Perhaps dividing this page
into four blocks would help.
(-1, -3) 7x + 2y = -13
8 – 4y = 21 + x
a.
The given point
is
/
is not a solution of the system of equations.
b.
The given point
is
/
is not a solution of the system of equations.
c.
The given point
is
/
is not a solution of the system of equations.
d.
The given point
is
/
is not a solution of the system of equations.
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2. Carefully graph the system of equations on the axes below.
Label your graph correctly.
a) x + y = 0
-x + y = -2
b) Where do the graphs intersect? Write your
answer as a solution set.
Although there are clues
given about where the axes
might be placed, it is not
clear whether they must be
placed using the clues.
The solution is ___________
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3. Use a graph to solve the system of equations. Label all
axes.
x – y = -3
2x + 3y = -6
The solution is ___________
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4. State whether each system is consistent and independent,
consistent and dependent or inconsistent.
a.
b.
y
y
Both vocabulary and
concepts must be wellunderstood for students to
respond to these items.
y = 3x + 2
y = 3x -2
x
x
3x +3y = 0
y = -x
The solution is ________________
c.
The solution is ________________
d.
y
y
X=1
y= ½ x
y = -x + 3
x
(2,1)
y = -2
x
The solution is ________________
e.
The solution is ________________
f.
y
y
y = 0.5x + 1
x = -2
2x – 4y = -4
x=3
x
x
The solution is ________________
The solution is ________________
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5. Use the substitution method to solve the following system of
equations. Show your work below and check your answer:
x–y= -3
2x + 3y = -6
McCymonds Algebra 1: Systems of Equations
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6. Use the elimination method to solve the following systems
of equations. Show your work below and check your
answer:
a. 5x + 3y = 17
5x – 2y = -3
b. 5b + 10c = 15
3b – 2c = -7
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7. Solve two of the four systems of equations using a different
method for each. Explain why you used each method.
Show your work on the pages provided and check your
answers.
a.
x + 3y = 0
b.
x + 4y = 10
c. 3x – 2y = 10
x + y = 12
1/2x + 1/4y = 4
d.
5x + 3y = 4
Recognizing why a
particular method may be
more appropriate for
solving a particular system
of equations is an important
component of the standard.
Allowing students to
choose which items to
complete means that they
must evaluate each of the
problems before making
their choices. This optional
feature is usually welcomed
by students.
x – y = -1
2x + y = 4
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8. Paul has 25 dimes and nickels. He has a total of $2.00.
How many dimes and nickels does Paul have? Show all
work below and check your answer.
The word problems are
important for assessing
applications of the standard
but the language may
prove to be a barrier for
English language learners.
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9. The sum of the two digits of a two-digit number is 10. If
you reverse the digits, the new number is one less than
twice the original number. Find the original number.
Show all work below and check your answer.
McCymonds Algebra 1: Systems of Equations
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10. State the equations you would use to solve this problem.
Then solve. Show all work below and check your answer.
Tickets to McClymonds High School Concert, featuring
Destiny’s Child Cost $11.00 for students and $4.00 for
faculty. On Tuesday 36 tickets were sold for a total of
$214.00. How many tickets of each kind where sold?
The solution is ___________
McCymonds Algebra 1: Systems of Equations
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IV. CRITERIA FOR SUCCESS
What will be expected of the students on the
assessment
Characteristics of a High Quality Response to the
Assessment:
G. Correctly graph a set of equations on the same Cartesian
coordinate system
1. Set up and label coordinate system
2. Provide appropriate scale
5. Represent given equation graphically, accurately and
neatly
6. Plot points accurately and neatly
The specificity of the
characteristics makes it
clear what is required in
relation to the assessment.
H. Identify the solution set from the graph as an ordered pair
2. Explain clearly parallel, identical and intersecting lines
I.
Solve systems of linear equations using substitution.
J.
Solve systems of linear equations using elimination/linear
combination
K. Identify the most appropriate method for solving a system
L. Solve application problems involving a system of linear
equations
4. Accurately identify the variables before setting up the
problem and translating the word problem into an equation.
5. Correctly translate word problems into appropriate
equations.
6. Solve the system accurately using the most appropriate
method and labeling the answer using the correct units.
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V.
OPPORTUNITIES TO LEARN AND PERFORM
Instructional plan to assure that every student has
adequate opportunities to learn and practice what is
expected.
Opportunities to Learn:
Students will learn:
To graphically represent simultaneous equations
To use substitution and elimination methods to solve
simultaneous equations
To identify the most appropriate method for solving systems of
equations
To identify systems of equations to application problems.
Opportunities to Perform:
Students will engage in activities where they will:
1. Graph linear equations, systems of equations
2. Identify solution sets for systems of equations from graphs
3. Identify systems of equations that are parallel/inconsistent,
perpendicular/consistent, dependent and independent
4. Identify solution sets
5. Interpret algebraic, graphic and verbal representations of
systems of equations
6. Solve systems of equation using substitution
7. Solve systems of equations using elimination
8. Identify systems of equations that are consistent, inconsistent,
dependent and independent
9. Graph solution sets after solving systems of equations
algebraically
10. Students will use large chart paper to reproduce o the graphs
for display
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VI. THE PERFORMANCE STANDARD
Rubric or other form of scoring guides
The Assessment is scored on a Four – Point Rubric
Level 4 –
This response offers clear and convincing evidence of a
good knowledge of the mathematics related to the
standard being assessed.
Characteristics:
- Student work demonstrates all the characteristics of a
high quality response, specifically graphical
representations of linear equations in one and two
variables.
- Student solved systems of equations by methods of
elimination and by substitution.
- Student work is presented clearly and logically.
- Student checks all solutions.
- Students paid attention to all detail.
- Student followed all directions.
- Student displayed competence in all areas being
assessed.
The qualitative descriptions
of the levels are
informative. The
characteristics provide
more quantitative
description and are aligned
with the definition of the
standard but may require
some refinement in terms
of qualifiers such as “most”
and “some.”
Level 3 –
This response offers evidence of substantial knowledge
of the mathematics related to the standard being assessed.
Characteristics:
- Student demonstrates most of the characteristics of a
high quality response including graphical
representations.
- Student solved systems of equations by method of
substitution and elimination.
- Student work is presented with little errors.
- Student checked most of the solutions.
- Students paid attention to most of the instructions.
- Student followed most of the directions.
- Student displayed competence in most areas being
assessed.
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Level 2This response offers limited or inconsistent evidence of
knowledge of the mathematics related to the standard
being assessed.
Characteristics:
- Student work is presented with many errors.
- Student checked a few or none of the problems.
- Student did not pay attention to all of the instructions.
- Student work did not show quantitative relationships
graphically.
- Student work showed the application of the use of
only one method in solving systems of linear
equations.
Level 1 –
This response offers little or no evidence of knowledge of
the mathematics related to the standard being assessed.
Characteristics:
- Student showed inability to demonstrate solving one
and two step linear equations
- Student showed inability to use information from a
graph or equation to answer a problem situation.
- Student showed inability to identify ordered pairs of
data from a graph and interpret the meaning of the
data in terms of the situation depicted by the graph.
- Student demonstrated major mathematical errors.
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VIII. SAMPLES OF STUDENT WORK WITH
COMMENTARY
Commentary – Overview:
The samples of Student Work that follow are based on an earlier
version of the assessment. Only the prompts were changed in the
later version; the math problems are the same in both versions.
It is difficult to know if
samples of different pages
are from same or different
students.
Without complete or similar
samples of student work it
is difficult to really see the
differences between the
levels of achievement.
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MEETS THE STANDARD:
Performance Level 4
Commentary
Student identified
each problem clearly
and showed work,
even though a sign
error occurred when
solving one of the 4
problems
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MEETS THE STANDARD:
Performance Level 4 cont.
Commentary
Student demonstrated
a good response by
clearly showing the
work
•
set up and labeled
coordinate system.
•
provided
appropriate scale
•
represented the
equation
graphically
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MEETS THE STANDARD:
Performance Level 3
Commentary
Solution was correct but
checking was not done.
In using the elimination
method, it was not
clearly indicated which
equation is to be
multiplied by a factor.
a) 5X+3Y=17
5X-2Y=-3
(X) by -1
b) 5B+10C=15
3B-2C=-7
(X) by -1
Student used elimination
method successfully to
solve both problems.
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MEETS THE STANDARD:
Performance Level 3 cont.
Commentary
Student had correct set
up of the system of
equations. There was
inconsistency in the
solution.
i.e., 10d+5n=200 but
worked with
10(n+25)+5=2.0
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DOES NOT YET MEET THE STANDARD:
Performance Level 2
Commentary
Student had a problem
with signs, failed to
check solution to see if it
satisfied the systems of
equations.
The problem was one of
carelessness in combining
5b and 15b. All other
procedures appear to be
correct. The student
appears to be able to use
elimination as a method for
solving systems of
equations.
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DOES NOT YET MEET THE STANDARD:
Performance Level 2 cont.
Commentary
Student did not have a
concept of money and
the fact that the number
of coins should be a
whole number.
The student also did not
label the variables he/she
used. This is an indication
that reteaching about
methods for solving word
problems may be
warranted.
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DOES NOT YET MEET THE STANDARD:
Performance Level 1
Commentary
Student was unable to
solve one and two step
linear equation where
substituting the value
for x and solving for y
could not be done..
This student made an
arithmetic error and did not
bother to check the values
in the T-table. This led to
incorrect graphs and the
inability to solve the
problem.
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DOES NOT YET MEET THE STANDARD:
Performance Level 1 cont.
Commentary
Student had a limited
grasp of the prerequisite
skills that lead to the
standard.
Without seeing other
samples of this student’s
work, it is difficult to see
what the level of
competency is or whether
decoding the problem is
presenting the difficulty.
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Student Assessment
McCymonds Algebra 1: Systems of Equations
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Assessment
California Content Standards
Mathematics
SYSTEMS OF EQUATIONS
Algebra I
Standard 9
Students solve a system of two linear equations in two variables algebraically and are able
to interpret their answers graphically.
Directions: Read each problem carefully and show all steps that lead to your
solution.
Write your answers in the space provided.
1. Determine whether the given point is a solution of the system of equations. Show
your work.
a)
(2,8)
y = 3x + 2
b)
y = 9x - 10
c)
(-4,6) y = x – 2
y = -7x + 22
(10, -2) 7y + x = 14
d)
x = 10
(-1, -3) 7x + 2y = -13
8 – 4y = 21 + x
a. The given point
is
/ is not
a solution of the system of equations.
b. The given point
is
/ is not
a solution of the system of equations.
c. The given point
is
/ is not
a solution of the system of equations.
d. The given point
is
/ is not
a solution of the system of equations.
2. Carefully graph the system of equations on the axes below. Label your graph
correctly.
a)
x+y=0
-x + y = -2
b)
Where do the graphs intersect? Write your answer as a
solution set.
The solution is ___________
3. Use a graph to solve the system of equations. Label all axes.
x – y = -3
2x + 3y = -6
The solution is ___________
4. State whether each system is consistent and independent, consistent and dependent
or inconsistent.
a.
b.
y
y
y = 3x + 2
y = 3x -2
x
x
3x +3y = 0
y = -x
The solution is ________________
The solution is ________________
d.
d.
y
y
X=1
y= ½ x
y = -x + 3
x
(2,1)
x
y = -2
The solution is ________________
The solution is ________________
e.
f.
y
y
x = -2
y = 0.5x + 1
x=3
2x – 4y = -4
x
x
The solution is ________________
The solution is ________________
5. Use the substitution method to solve the following system of equations. Show your
work below and check your answer:
x–y= -3
2x + 3y = -6
6. Use the elimination method to solve the following systems of equations. Show your
work below and check your answer:
a. 5x + 3y = 17
5x – 2y = -3
b. 5b + 10c = 15
3b – 2c = -7
7. Solve two of the four systems of equations using a different method for each.
Explain why you used each method. Show your work on the pages provided and
check your answers.
a.
x + 3y = 0
x + 4y = 10
c. 3x – 2y = 10
5x + 3y = 4
b.
x + y = 12
1/2x + 1/4y = 4
d. x – y = -1
2x + y = 4
8. Paul has 25 dimes and nickels. He has a total of $2.00. How many dimes and
nickels does Paul have? Show all work below and check your answer.
9. The sum of the two digits of a two-digit number is 10. If you reverse the digits, the
new number is one less than twice the original number. Find the original number.
Show all work below and check your answer.
10. State the equations you would use to solve this problem. Then solve. Show all work
below and check your answer.
Tickets to McClymonds High School Concert, featuring Destiny’s Child Cost $11.00 for
students and $4.00 for faculty. On Tuesday 36 tickets were sold for a total of $214.00.
How many tickets of each kind where sold?
Instructional Leadership Initiative: Supporting Standards-based Practice
Answer Key
McCymonds Algebra 1: Systems of Equations
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Algebra Assessment: Answer Key
1.
Determine whether the given point is a solution of the system of equations. Show your
work.
a)
(2,8)
y = 3x + 2
b)
(-4,6) y = x – 2
y = 9x - 10
c)
(10, -2) 7y + x = 14
y = -7x + 22
d)
(-1, -3) 7x + 2y = -13
x = 10
a)
b)
c)
8 – 4y = 21 + x
8 = 3(2) + 2
8 = 6 + 2√
8 = 9(2) – 10
8 = 18 – 10√
6 = (-4) – 2
6 ≠ -6
6 = -7(-4) + 22
6 ≠ 28 + 22
7(-2) + 10 = 14
-14 + 10 ≠ 14
d) 7(-1) + 2(-3) = -13
-7 + (-6) = -13√
Yes [2/2 pts]
No [2/2 pts]
10 = 10√
No [2/2 pts]
8 – 4(-3) = 21 + (-1)
8 + 12 = 20√
Yes [2/2 pts]
a.
The given point
is
/ is not a solution of the system of equations.
[.5/.5 pt]
b.
The given point
is
/ is not a solution of the system of equations.
[.5/.5 pt]
c.
The given point
is
/ is not a solution of the system of equations.
[.5/.5 pt]
d.
The given point
is
/ is not a solution of the system of equations.
[.5/.5 pt]
Algebra Assessment: Answer Key
2.
Carefully graph the system of equations on the axes below. Label your graph correctly.
a)
x+y=0
-x + y = -2
b)
Where do the graphs intersect? Write your answer as a solution set.
x+y=0
x
0
2
-2
y
0
-2
2
-x + y = -2
x
0
2
y
-2
0
[2/2 pts]
[2/2 pts]
y
intersection point
(1, -1)
x+y=0
-x + y = -2
2 points per correct point
and line [4/4 pts]
x
1 pt intersection
1 pt labeling
The solution is ___(1,
-1)____
Algebra Assessment: Answer Key
3.
Use a graph to solve the system of equations. Label all axes.
x – y = -3
2x + 3y = -6
x – y = -3
x
0
-3
2x +3y = -6
y
3
0
x
-3
0
[2/2 pts]
y
0
-2
[2/2 pts]
y
x - y = -3
2 points per correct point and line
[4/4 pts]
x
2x + 3y = -6
1 pt intersection
1 pt labeling
The solution is _(-3,
0)____
Algebra Assessment: Answer Key
4.
State whether each system is consistent and independent, consistent and dependent or
inconsistent.
[2 points for each correct answer; 10 points for entire page]
a.
b.
y
y
y = 3x + 2
y = 3x -2
x
x
3x +3y = 0
y = -x
The solution is _consistent, dependent
The solution is _inconsistent______
e.
d.
y
y
X=1
y= ½ x
y = -x + 3
x
(2,1)
x
y = -2
The solution is _consistent, independent
The solution is _consistent, independent
e.
f.
y
y
x = -2
y = 0.5x + 1
2x – 4y = -4
x=3
x
x
The solution is _consistent, dependent
The solution is _inconsistent______
Algebra Assessment: Answer Key
5.
Use the substitution method to solve the following system of equations. Show your work
below and check your answer:
x–y= -3
2x + 3y = -6
x=y–3
2(y – 3) + 3y = -6
2y – 6 + 3y = -6
5y = 0
y=0
x = -3
(-3,0)
-3 – 0 = -3 √
2(-3) + 3(0) = -6
-6 = -6 √
4 pts = find x
4 pts = find y
2pts = check solution
Algebra Assessment: Answer Key
6.
a.
Use the elimination method to solve the following systems of equations. Show your work
below and check your answer:
5x + 3y = 17
-(5x – 2y
= -3)
5y = 20
y = 4 [2/2 pts]
5x + 12 = 17
-12 = -12
5x = 5
x=1
(1,4) [2/2 pts]
5(1) + 3(4) = 17
5 + 12 = 17√
[1/1 pt]
5(1) + -2(4) = -3
5 + -8 = -3√
b.
5b + 10c = 15
5(3b – 2c
= -7)
5B + 10C = 15
15B – 10C = -35
20B = -20
B = -1 [2/2 pts]
-5 + 10C = 15
+5 +5
10C = 20
C=2
(-1, 2) [2/2 pts]
5(-1) + 10(2) = 15
-5 + 20 = 15√
[1/1 pt]
3(-1) – 2(2) = -7
-3 – 4 = -7 √
Algebra Assessment: Answer Key
7.
Solve two of the four systems of equations using a different method for each. Explain why
you used each method. Show your work on the pages provided and check your answers.
(5 pts for each correct solution, 10 pts for entire page)
a.
x + 3y = 0
-(x + 4y = 10)
b.
x + y = 12
1/2x + 1/4y = 4
 x 12 − x

 2 + 4 = 4
-y = -10
4
y = 10
x + 30 = 0
x = -30
(-30, 10)
c)
(3x – 2y = 10)3
(5x + 3y = 4)2
9x - 6y = 30
10x +6y = 8
19x = 38
19 19
x = 38 = 2
19
6 – 2y = 10
-6
= -6
-2y = 4
y = -2
(2,-2)
2x + 12 – x = 16
-12 = -12
x=4
y=8
(4,8)
d.
x – y = -1
2x + y = 4
3x = 3
x=1
1 – y = -1
-1 = -1
-y = -2
y=2
(1,2)
Algebra Assessment: Answer Key
8.
Paul has 25 dimes and nickels. He has a total of $2.00. How many dimes and nickels does
Paul have? Show all work below and check your answer.
n + d = 25 [3/3 pts]
.05n + .10d = 2.00 [3/3 pts]
n = 25 – d
n + d = 25
5n + 10d = 200
5(25 – d) + 10d = 200
125 – 5d + 10d = 200
125 + 5d = 200
-125
-125
5d = 75
d = 15 [2/2 pts]
n = 25 – 15
n = 10 [2/2 pts]
Algebra Assessment: Answer Key
9.
The sum of the two digits of a two-digit number is 10. If you reverse the digits, the new
number is one less than twice the original number. Find the original number. Show all
work below and check your answer.
let x be the unit digit.
let y be the tens digit
So
Here
x + y = 10 . . [2/2 pts]
x = 10 – y
Reversing the digits
10x + y + 1 = 2 (10y + x)
10x + y + 1 = 20y + 2x
8x – 19y = -1
8(10-y) – 19y = -1
80 – 8y – 19y = -1
-27y = -81
y = 3 [2/2 pts]
From equation (#1)
x = 10 – 3
x = 7 [2/2 pts]
The number is 37
[2/2 pts]
Check
73 + 1 = 2(37)
74 = 74
Algebra Assessment: Answer Key
10.
State the equations you would use to solve this problem. Then solve. Show all work below
and check your answer.
Tickets to McClymonds High School Concert, featuring Destiny’s Child Cost $11.00 for
students and $4.00 for faculty. On Tuesday 36 tickets were sold for a total of $214.00. How
many tickets of each kind where sold?
s + f = 36 … s =36 – f [3/3 pts]
11s + 4f = 214
[3/3 pts]
11(36 – f) + 4f = 214
396 – 11f + 4f = 214
396 – 7f = 214
-396
-396 [2/2 pts]
-7f = -182
f = 26
s = 10 [2/2 pts]
26 faculty tickets and 10 student tickets were sold.