Systems of Linear Equations by Michael Eber
Standards Addressed:
M8A4. Students will graph and analyze graphs of linear equations and inequalitites.
a. Interpret slope as a rate of change.
b. Determine the meaning of the slope and y-intercept in a given situation.
c. Graph equations of the form y = mx + b.
d. Graph equations of the form ax + by = c.
M8A5. Students will understand systems of linear equations and inequalities and use
them to solve problems.
a. Given a problem context, write an appropriate system of linear equations or
inequalities.
b. Solve systems of equations graphically and algebraically, using technology as
appropriate.
d. Interpret solutions in problem contexts.
M8P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M8P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
Question:
Why can’t you solve every system of equations using graphing?
Launch
A solution to a system of linear equations is the point that makes both equations true. If
you have graphed the equations, the solution is found at the point of intersection of each
graph. Every system can also be solved using the graphing method, substitution method,
or linear combination (elimination) method.
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For the following system of equations prove that each method of solving comes up with
the same solution:
y+x=5
and
y–x=3
Investigation:
Part 1—Finding a solution graphically
In the Launch, you found that a solution to a system of equations should be the same
regardless of the method used to solve the system. We are going to look at a specific
problem where the solutions might not be the same regardless of the method used.
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The Problem
Marcy and Kwaneisha are both walking in the woods outside of their town.
Marcy starts walking in a straight line where for every mile she walks east she walks 5
miles north. Kwaneisha also walks in a straight line but she begins walking from a point
7 miles north of Marcy’s starting point. She follows a line where for every 3 miles east
she walks she goes 1 mile south.
a. If you placed both Marcy’s and Kwaneisha’s path on the same graph, what
would each of their y-intercepts be?
b. What would the slope of each girl’s path be?
c. Will their paths intersect? Explain how you know this to be true?
d. What is their point of intersection?—show this by graphing both girl’s path
and identifying the point of intersection.
Part 2 − Finding a solution algebraically.
a. Using the problem from part A above—what is the equation of Marcy’s path?
b. What is the equation of Kwaneisha’s path?
c. Solve the system of equations using the substitution method.
d. Solve the system of equations using the linear combination or elimination
method.
e. Are the solutions the same in parts c and d?
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f. Are these two solutions the same as the solution you got graphically?
Conclusions:
Write a statement explaining how to find each of the following:
1. When is the solution to a set of equations the same regardless of the method you
used?
2. When does the graphing method give you a different solution to a set of linear
equations than the algebraic methods?
In Class Problems:
Break into groups of 3 to work on these 3 problems. For each problem, pick 1
method per a person (graphing, linear combination, and substitution) and individually
solve the problem using that method—then compare your answers.
1. y= 2/3x + 5
y = -3/2 x + 5
a. Solve the problem using the method you chose.
b. Compare your answer within your group—did each person get the same answer?
If not, explain why.
c. Decide as a group what was the best method to solve this problem—defend your
choice of the best method.
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2. y = x – 8
2x + 4y = 7
a. Solve the problem using the method you chose.
b. Compare your answer within your group—did each person get the same answer?
If not, explain why.
c. Decide as a group what the best method to solve this problem was—defend your
choice of the best method.
3. 3x + 4y = 8
5x – 8y = 12
a. Solve the problem using the method you chose.
b. Compare your answer within your group—did each person get the same answer?
If not, explain why.
c. Decide as a group what was the best method to solve this problem—defend your
choice of the best method.
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Closure:
1. Why can’t you solve every system of equations using graphing? Justify your
answer.
2. How can you determine which method is the best method for solving a system of
linear equations? Explain your reasoning.
Homework:
For each problem (#1-6), solve using the method of your choice. Explain why
you chose that method and why it was the best method for you on that problem.
1. 2x – 8y = 21
x = 4y
2. y = 4
3. 3x + 2y = 7
5x + 7y = 11
4. x – y = -3
y = 6x -8
2y – 6x = 4
5. Mary went to the store to buy some clothes. She was looking to buy shorts and a
shirt. The store only showed two specials on shorts and shirts. The first one was buy 2
pairs of shorts and 3 shirts for $141.50. The second deal is buy 5 pairs of shorts and 4
shirts for $269.75. How much would 4 pairs of shorts and 2 shirts cost?
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6. Terrance and Jose had to do sprints after football practice for mistakes they made.
Together they did 49 sprints. If Terrance did four more than five times Jose. How many
sprints did each of them do after practice?
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