Focus on Math Concepts
Lesson 15
Part 1: Introduction
CCSS
8.EE.C.8a
Understand Systems of Equations
How can graphs help you find the solution of two linear equations?
In Lesson 14, you reviewed the structure of linear equations that have no solution, one
solution, and infinitely many solutions. Now you will examine the graphs of pairs of linear
equations to see if both equations have a common solution.
A system of linear equations is two or more related linear equations for which you are
trying to find a common solution. The examples you will work within this lesson have two
equations and two variables. To solve a system of equations, you need to find the ordered
pair (or pairs) that solves both equations in the system. One way to find the solution is to
graph the equations on the same coordinate plane.
Think Graphs of two linear equations might have no point of intersection.
Look at a graph of this system of equations.
y 5 3x
Circle the two variables
that are used in this
system of equations.
y 5 3x 2 2
4
3
2
1
y
24232221 0 1 2 3 4 x
21
22
23
24
These lines don’t intersect, so there is no ordered pair that will satisfy both equations.
There is no point that lies on both lines. Like an equation that has no solution, this system
has no solution.
132
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Part 1: Introduction
Lesson 15
Think Graphs of two linear equations might intersect in exactly one point.
Here is a graph of a system of equations with exactly one solution.
The point where the lines intersect, (0, 3), is a solution to both equations, so it solves the
system. Like an equation that has exactly one solution, this system has exactly one solution.
You can substitute (0, 3) into both equations to verify that it is a solution to both.
y 5 22x + 3
3 5 22(0) + 3
353
y5x13
35013
353
y 5 22x 1 3 and y 5 x 1 3
5
4
3
2
1
y
24232221 0 1 2 3 4 x
21
22
23
Think Graphs of two linear equations might intersect at all points.
Here is a graph of a system of equations with infinitely many solutions.
These two equations produce the same set of ordered pairs. When you substitute any value
for x into each equation, you get the same y-values for each.
y 1 2 5 2x and y 5 2(x 2 1)
y 1 2 5 2x
x 22 0 2
y 26 22 2
4
6
y 5 2(x 2 1)
x 22 0 2 4
y 26 22 4 6
8
6
4
2
y
24232221 0 1 2 3 4 x
22
24
26
28
Reflect
1 Describe the number of solutions to a system of linear equations when the graph of the
equations do not intersect, intersect in exactly one point, and intersect at all points.
L15: Understand Systems of Equations
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133
Part 2: Guided Instruction
Lesson 15
Explore It
Examining graphs and equations can help you determine if a system of linear
equations has no solution.
y
2 What are the slope and y-intercept of line r and line s?
4
3
2
s
1
3 Look back at the graph with no solution in the introduction.
-4 -3 -2 -1 0
-1
1 2
-2
Are the slopes for each line the same or different? Are the
y-intercepts the same or different?
-3
3 4
x
r
-4
4 Why will these pairs of lines never intersect?
5 When does a system of linear equations have no solution? Explain.
Examining graphs and equations can help you determine if a system of linear
equations has infinitely many solutions.
y
6 Substitute 4 and then 24 for x in the equations
y 5 x 1 1 and 2y 2 2 5 2x. Describe the results.
7 Divide each term in 2y 2 2 5 2x by 2; then write it in
slope-intercept form.
8 Now compare the equation you wrote in problem 7
with the first equation in problem 6.
4
3
2
1
24232221 0 1 2 3 4
21
22
23
24
x
9 Find the system of equations in the introduction that has infinitely many solutions. Write
each equation in slope-intercept form. Describe the results.
134
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Part 2: Guided Instruction
Lesson 15
Talk About It
Solve the problems below as a group.
10 Look back at the system of equations in the introduction that has exactly one solution.
Are the slopes of the two equations the same or different? Are the y-intercepts the same
or different?
11 Compare the slopes and y-intercepts for the equations in
this system of equations, and then graph the equations.
How many solutions does this system of equations have?
y5x25
y 5 2x 2 5
y
7
6
5
4
3
2
1
27262524232221 0
21
22
23
24
25
26
27
1 2 3 4 5 6 7
x
12 Each situation below describes a possible relationship between the equations in a
system of equations. Write the number of solutions there are for each situation.
Equations with same slope and same y-intercept:
Equations with same slope and different y-intercepts:
Equations with different slopes and same y-intercept:
Equations with different slopes and different y-intercepts:
Try It Another Way
Examine each system of equations. Do both equations have the same slope or
different slopes? The same y-intercept or different y-intercepts? Predict what kind of
solution each one has. Justify your answers.
13 y 1 1 5 1 x
3
··
y 5 1x
3
··
14 y 5 3x 2 1
y 2 2 5 1x
2
··
L15: Understand Systems of Equations
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135
Part 3: Guided Practice
Lesson 15
Connect It
Talk through these problems as a class, then write your answers below.
15 Analyze: Look at the system of equations below. Without graphing, explain how you can
tell what type of solution this system has.
y51x+3
2
··
y 5 0.25(2x + 4)
16 Evaluate: Look at the system of equations in the box at the right.
Kevin stated that the equations in this system have the same slope,
so there will be infinitely many solutions. Does Kevin’s statement
make sense? Explain.
17 Create: Study the graph at the right. Based on what
you see, write an equation for each line. Substitute
the coordinates of the point of intersection into both
of your equations. Explain why this point is a solution
of the system of equations.
System of Equations
y 5 1.5x 1 4
y 5 1.5x 2 2
4
3
2
1
y
24232221 0 1 2 3 4 x
21
22
23
24
136
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Part 4: Common Core Performance Task
Lesson 15
Put It Together
18 Use what you have learned to complete this task.
Create a system of equations with the following number of solutions. Use the
equation below as one of the equations in each system. Justify your answers.
y 5 1 (6x 2 8)
2
··
A exactly one solution
8
7
6
5
4
3
2
1
21
21
22
23
y
0
1
2
3
4 x
B infinitely many solutions
C no solution
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137
Part 1: Introduction
Lesson 15
AT A GLANCE
Focus on Math Concepts
Students explore the graph of a system of equations that
has no point of intersection and therefore no solution.
Lesson 15
Part 1: Introduction
CCSS
8.EE.C.8a
Understand Systems of Equations
STEP BY STEP
How can graphs help you find the solution of two linear equations?
• Introduce the Question at the top of the page.
In Lesson 14, you reviewed the structure of linear equations that have no solution, one
solution, and infinitely many solutions. Now you will examine the graphs of pairs of linear
equations to see if both equations have a common solution.
• Direct students’ attention to the fact that graphing
a system of linear equations lets them find whether
two linear equations have a common solution.
A system of linear equations is two or more related linear equations for which you are
trying to find a common solution. The examples you will work within this lesson have two
equations and two variables. To solve a system of equations, you need to find the ordered
pair (or pairs) that solves both equations in the system. One way to find the solution is to
graph the equations on the same coordinate plane.
Think Graphs of two linear equations might have no point of intersection.
• Read Think with students. Ask them to name the
variables used in the system of equations. [x and y]
y 5 3x
• Refer students to the graph. Lead a discussion
comparing the two equations. Listen for an
understanding that the equations share a slope but
have different y-intercepts, so they are parallel lines.
y 5 3x 2 2
4
3
2
1
y
24232221 0 1 2 3 4 x
21
22
23
24
• Explain that an ordered pair that “satisfies both
equations” in this situation means that the same
point will lie on both lines. Stress that because these
lines are parallel, they do not share a common point,
and therefore do not share a common solution.
ELL Support
Circle the two variables
that are used in this
system of equations.
Look at a graph of this system of equations.
These lines don’t intersect, so there is no ordered pair that will satisfy both equations.
There is no point that lies on both lines. Like an equation that has no solution, this system
has no solution.
132
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Explain that a system of equations is a set of related
equations. The variables in the equations represent
the same value or values: x in one equation has the
same value as any x in the other equations.
Hands-On Activity
Explore systems of equations graphically.
Materials: graph paper, ruler
Walk students through the following steps:
• Work with a partner to graph these systems:
System 1
System 2
y 5 3x 1 2
y 5 22x 2 4
y5x12
y 5 22x 2 7
• Determine if the systems have no solution,
infinitely many solutions, or one solution.
146
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Part 1: Introduction
Lesson 15
AT A GLANCE
Students explore a system of equations with exactly
one solution. Then they explore a system with an
infinite number of solutions.
STEP BY STEP
• Ask students to describe in their own words what
they think the graph of a system with one solution
would look like. [The lines would intersect at a single
point, which represents the solution.]
• Read the first Think as a class. Explain that a
common point, or a point of intersection, is
considered a solution to a linear system.
• Direct students’ attention to the first graph. Ask a
volunteer to use the graph to name the point of
intersection, or solution, to the linear system. [0, 3]
• Have a volunteer come to the board to show how
the point of intersection can be used to verify the
solution. [Substitute the x- and y-coordinates of
the point of intersection into each linear equation.
Then solve each equation to verify that the results
are true mathematical statements.]
• Explain that it is possible for a linear system to have
infinitely many solutions. Ask students to make
predictions as to how this can happen.
• Read the second Think as a class, then direct
students’ attention to the graph. Ask, What is different
about this graph compared to the other two graphs
you’ve seen? [There is only one line in this graph. The
equations in the system produce the exact same line
on the graph.]
• Explain that when the equations in a system
represent the same line, the system has infinitely
many solutions.
• Have students read and reply to the Reflect directive.
SMP Tip: Students model with mathematics as they
analyze and discuss systems of equations
represented in a table, with equations, and on a
graph (SMP 4).
L15: Understand Systems of Equations
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Part 1: Introduction
Lesson 15
Think Graphs of two linear equations might intersect in exactly one point.
Here is a graph of a system of equations with exactly one solution.
The point where the lines intersect, (0, 3), is a solution to both equations, so it solves the
system. Like an equation that has exactly one solution, this system has exactly one solution.
You can substitute (0, 3) into both equations to verify that it is a solution to both.
y 5 22x + 3
3 5 22(0) + 3
353
y5x13
35013
353
y 5 22x 1 3 and y 5 x 1 3
5
4
3
2
1
y
24232221 0 1 2 3 4 x
21
22
23
Think Graphs of two linear equations might intersect at all points.
Here is a graph of a system of equations with infinitely many solutions.
These two equations produce the same set of ordered pairs. When you substitute any value
for x into each equation, you get the same y-values for each.
y 1 2 5 2x and y 5 2(x 2 1)
y 1 2 5 2x
x 22 0 2
y 26 22 2
8
6
4
2
4
6
y 5 2(x 2 1)
x 22 0 2 4
y 26 22 4 6
y
24232221 0 1 2 3 4 x
22
24
26
28
Reflect
1 Describe the number of solutions to a system of linear equations when the graph of the
equations do not intersect, intersect in exactly one point, and intersect at all points.
When the lines do not intersect, the system has no solution. When the lines
intersect at one point, there is exactly one solution. When the lines intersect at
all points, there are infinitely many solutions.
133
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Mathematical Discourse
• How can you interpret the slope-intercept form of an
equation to determine whether a system has one
solution, no solution, or infinitely many solutions?
Equations with different slopes will have
exactly one solution. Equations with the same
slope but different y-intercepts will have no
solution. Equations that can be rewritten to
have the same slope and same y-intercept have
infinitely many solutions.
• Name a real-world example of a linear system that
has infinitely many solutions?
An example is the equations used to convert
temperatures in Celsius and Fahrenheit:
C 5 5 (F 2 32) and F 5 9 C 1 32. These
9
··
5
··
equations can be rewritten to be identical.
147
Part 2: Guided Instruction
Lesson 15
AT A GLANCE
Part 2: Guided Instruction
Students will examine a system of equations with no
solution and a different system of equations with an
infinite number of solutions.
Lesson 15
Explore It
Examining graphs and equations can help you determine if a system of linear
equations has no solution.
y
2 What are the slope and y-intercept of line r and line s?
Line r: slope is 2 1, y-intercept is 0.
3
··
Line s: slope is 2 1 , and y-intercept is 2.
3
··
STEP BY STEP
• Tell students they will have time to work individually
on the Explore It problems on this page and then
share their responses in groups. You may choose to
work through the first problem together as a class.
4
3
2
s
1
-4 -3 -2 -1 0
-1
3 Look back at the graph with no solution in the introduction.
1 2
-2
Are the slopes for each line the same or different? Are the
y-intercepts the same or different?
-3
3 4
x
r
-4
Slopes are the same, y-intercepts are different.
4 Why will these pairs of lines never intersect?
Each pair of lines will never intersect because they are parallel.
• As students work individually, circulate among them,
assessing student understanding and addressing
misconceptions. Use the Mathematical Discourse
questions to engage student thinking.
5 When does a system of linear equations have no solution? Explain.
Possible answer: If the two equations in a system have the same slope and
different y-intercepts, the lines are parallel, and there is no solution.
Examining graphs and equations can help you determine if a system of linear
equations has infinitely many solutions.
y
6 Substitute 4 and then 24 for x in the equations
• Students may find it helpful to refer back to previous
problems to make connections among slopes,
y-intercepts, and common solutions.
y 5 x 1 1 and 2y 2 2 5 2x. Describe the results.
4
3
2
1
Possible answer: When x 5 4, y 5 5 in both equations.
When x 5 24, y 5 23 in both equations.
24232221 0 1 2 3 4
21
22
23
24
7 Divide each term in 2y 2 2 5 2x by 2; then write it in
slope-intercept form. y 2 1 5 x; y 5 x 1 1
• Take note of students who are having difficulty. Wait
to see if their understanding progresses as they work
in their groups during the next part of the lesson.
8 Now compare the equation you wrote in problem 7
with the first equation in problem 6.
x
Both equations are the same.
9 Find the system of equations in the introduction that has infinitely many solutions. Write
each equation in slope-intercept form. Describe the results.
y 1 2 5 2x is y 5 2x 2 2, and y 5 2(x 2 1) is y 5 2x 2 2. They are the same equation.
STUDENT MISCONCEPTION ALERT: Some
students struggle identifying the slope and y-intercept.
Remind students that in the slope-intercept form,
y 5 mx 1 b, m is the slope and b is the y-intercept.
Hands-On Activity
Work together to create systems of equations.
Materials: graph paper, ruler, and index cards
bearing the following expressions:
x 2 y 5 3, 5x 1 y 5 23, 2x 2 y 5 5, 7x 1 y 5 25,
5x 2 y 5 5, y 5 4, 10x 2 2y 5 10, 2x 1 y 5 10
Divide students into pairs. Have each pair choose
two index cards, write the equations on graph paper,
and replace the cards. Walk students through the
following steps:
• Work with a partner to graph your system of
equations. Determine whether the system has one
solution, no solutions, or infinitely many
solutions, and write down your results.
134
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Mathematical Discourse
• Why is it important to identify the slopes and
y-intercepts when determining solutions of a linear
system?
The slopes and y-intercepts tell us whether the
lines intersect (one solution), are parallel
(no solution), or are the same (infinite
solutions).
• How can you know whether a point would satisfy a
system of equations if you knew the system had
infinite solutions?
If the point satisfies any one line in the system,
it will satisfy the entire system because the
equations are equivalent.
• Share only your equations with other groups. Can
they determine whether the system has a solution
just by looking at the equations? After they have
answered, show them your graph to verify.
148
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Part 2: Guided Instruction
Lesson 15
AT A GLANCE
Students will analyze systems of equations and estimate
solutions.
STEP BY STEP
• Organize students into pairs or groups. You may
choose to work through the first Talk About It
problem together as a class.
• Walk around to each group, listen to, and join in on
discussions at different points. Use the Mathematical
Discourse questions to help support or extend
students’ thinking.
• Direct each group’s attention to Try It Another Way.
Have a volunteer from each group come to the board
to explain the group’s solutions to problems 13 and 14.
• When sharing, emphasize that both algebraic and
geometric reasoning can be used to analyze systems
of equations.
SMP Tip: Ask students to support their
explanations using both algebraic and geometric
reasoning. This provides an opportunity for
students to practice critiquing the reasoning of
others (SMP 3), perhaps by rephrasing, asking for
clarification, or identifying a misconception.
Conceptual Extension
Part 2: Guided Instruction
Lesson 15
Talk About It
Solve the problems below as a group.
10 Look back at the system of equations in the introduction that has exactly one solution.
Are the slopes of the two equations the same or different? Are the y-intercepts the same
or different? The slopes are different, and the y-intercepts are different.
y
11 Compare the slopes and y-intercepts for the equations in
7
6
5
4
3
2
1
this system of equations, and then graph the equations.
How many solutions does this system of equations have?
y5x25
y 5 2x 2 5
Possible answer:
The equations have a different slope but the same
y-intercept. There is exactly one solution: the y-intercept.
27262524232221 0
21
22
23
24
25
26
27
1 2 3 4 5 6 7
x
12 Each situation below describes a possible relationship between the equations in a
system of equations. Write the number of solutions there are for each situation.
Equations with same slope and same y-intercept: infinitely many
no solution
Equations with same slope and different y-intercepts:
exactly 1
Equations with different slopes and same y-intercept:
exactly 1
Equations with different slopes and different y-intercepts:
Try It Another Way
Examine each system of equations. Do both equations have the same slope or
different slopes? The same y-intercept or different y-intercepts? Predict what kind of
solution each one has. Justify your answers.
Possible answer: The coefficients of x are both 1, so the slopes are the
13 y 1 1 5 1 x
3
··
3
··
y 5 1x
3
··
14 y 5 3x 2 1
y 2 2 5 1x
2
··
same. Because the y-intercepts are different, there is no solution.
Possible answer: y 5 3x 2 1 and y 5 1x 1 2 have different slopes and
2
··
y-intercepts. They are not parallel, and they are not the same
equation. There is exactly one solution.
L15: Understand Systems of Equations
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135
Mathematical Discourse
Materials: Internet, books, graphing paper
• Identify two ways to determine solutions for a given
systems of equations.
You can solve a system algebraically and
geometrically.
• With a partner, use the Internet or books to write
a real-world problem that can be represented with
a system of equations. Some examples are systems
depicting temperature conversions, money
exchange rates, and cell-phone plan rates.
• Is it possible for three equations to have a common
solution? If not, explain your reasoning.
Yes, it is possible. Three equations could
produce three lines with unique slopes that all
intersect at one point.
Write real-world problems that can be
represented by systems of equations.
• Graph and analyze the system of equations to
determine the solution(s).
• Swap stories with another pairing. Can they
determine the solution(s)?
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149
Part 3: Guided Practice
Lesson 15
AT A GLANCE
Part 3: Guided Practice
Students demonstrate their understanding of solutions
to systems of equations.
Lesson 15
Connect It
Talk through these problems as a class, then write your answers below.
STEP BY STEP
15 Analyze: Look at the system of equations below. Without graphing, explain how you can
tell what type of solution this system has.
y51x+3
• Discuss each Connect It problem as a class using the
discussion points outlined below.
2
··
y 5 0.25(2x + 4)
Possible answer: Expand y 5 0.25(2x 1 4): y 5 0.5x 1 1. Compare y 5 0.5x 1 1 with
y 5 1 x 1 3. These equations have the same slope but different y-intercepts. That
2
··
means their graphs are parallel lines and there is no solution.
Analyze:
16 Evaluate: Look at the system of equations in the box at the right.
Kevin stated that the equations in this system have the same slope,
so there will be infinitely many solutions. Does Kevin’s statement
make sense? Explain.
• Have students graph the system of equations to
check their solutions.
System of Equations
y 5 1.5x 1 4
y 5 1.5x 2 2
Possible answer: In order to have infinitely many solutions, the equations must
be the same. These equations have the same slope but different y-intercepts.
• Ask students to discuss their reasoning, using their
graphs as supporting evidence.
Also, it’s not possible to add 4 to 1.5x and subtract 2 from 1.5x and get the same
answer. There is no solution.
17 Create: Study the graph at the right. Based on what
Evaluate:
and y 5 1x 23. The point of intersection is (4, 22).
4
··
y 5 2x 1 2
• Students may choose to support their explanations
with a graph.
y
24232221 0 1 2 3 4 x
21
22
23
24
Possible answer: The equations are y 5 2x 1 2
• Ask students to work in groups and share their
reasoning.
• Challenge students to change one of the equations in
such a way as to make Kevin’s statement true.
4
3
2
1
you see, write an equation for each line. Substitute
the coordinates of the point of intersection into both
of your equations. Explain why this point is a solution
of the system of equations.
y 5 1x 2 3
4
··
1 (4) 2 3
22 5 24 1 2
22 5
22 5 22
22 5 22
4
··
(4, 22) solves both equations, so it is a solution to the system of equations.
136
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Create:
• Have students work in groups and share their
solutions.
• Ask, Why is it important to check your answer from the
graph in the equations?
150
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Part 4: Common Core Performance Task
Lesson 15
AT A GLANCE
Part 4: Common Core Performance Task
Students will demonstrate their understanding of
systems of equations to solve a task.
Lesson 15
Put It Together
18 Use what you have learned to complete this task.
STEP BY STEP
Create a system of equations with the following number of solutions. Use the
equation below as one of the equations in each system. Justify your answers.
• Direct students to complete the Put It Together
task on their own.
y 5 1 (6x 2 8)
2
··
A exactly one solution
8
7
6
5
4
3
2
1
Possible answer: y 5 x 1 2. The graph
• As students work on their own, walk around to
assess their progress and understanding, to
answer their questions, and to give additional
support, if needed.
shows that the solution of the system of
equations is exactly one point. Both
equations have a different slope and
different y-intercept.
22
21
• If time permits, have students share their
reasoning for each solution.
21
22
23
y
0
1
2
3
4 x
B infinitely many solutions
Possible answer: I can expand the equation y 5 1 (6x 2 8): y 5 3x 2 4. The original
2
··
equation and the expanded one are exactly the same. The graph is the same line,
SCORING RUBRICS
so there are infinitely many solutions for y 5 1 (6x 2 8) and y 5 3x 2 4.
2
··
C no solution
See student facsimile page for possible student work.
Possible answer: I found in B that the equation has a slope of 3 and a y-intercept
A
would be a line parallel to the given line, and there would be no solution. The
of 24. The graph of an equation with the same slope and a different y-intercept
Points Expectations
2
1
0
B
system of equations y 5 3x 1 1 and y 5 1(6x 2 8) would result in no solution.
The response demonstrates the student’s
understanding of systems of linear
equations with exactly one solution. The
equation found produces a line that
intersects once with the given line.
An effort was made to accomplish the task.
The response demonstrates evidence of
understanding systems of linear equations
with one solution, but the reasoning may
contain some misunderstandings.
There is no response, or the response shows
little or no understanding of the task.
Points Expectations
2
The response demonstrates the student’s
understanding of systems of linear equations
with infinitely many solutions. The equation
found is mathematically equivalent to the
given equation.
1
An effort was made to accomplish the task.
The response demonstrates evidence of
understanding linear equations with
infinitely many solutions, but the reasoning
may contain some misunderstandings.
0
There is no response, or the response shows
little or no understanding of the task.
L15: Understand Systems of Equations
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2
··
L15: Understand Systems of Equations
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C
137
Points Expectations
2
The response demonstrates the student’s
understanding of systems of linear equations
with no solution. The equation found has the
same slope as, but a different y-intercept
than, the given equation.
1
An effort was made to accomplish the task.
The response demonstrates evidence of
understanding systems of linear equations
with no solution, but the reasoning may
contain some misunderstandings.
0
There is no response, or the response shows
little or no understanding of the task.
151