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Solving Systems of Equations

Lesson # 13

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2019 Open Up Resources |

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Let’s solve systems of equations.

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Today’s Goals

I can graph a system of equations.
I can solve systems of equations using algebra.

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True or False: Two Lines

Warm Up

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Use the lines to decide whether each statement is true or false. Be prepared to explain your reasoning using the lines.

Work on your own (2 min)
Discuss your answers with a partner (1 min)

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Matching Graphs to Systems

Activity 1

MLR8: Discussion Supports

Launch

Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and encourage them to do so.

If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so the students have another way of checking their solutions.

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Work on this question only on your own (5-7 min).��Pause when you are done.

Find the solution to each system and then check that your solution is reasonable on the graph.

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Let’s Talk About It

Do you need to see the graphs of the equations in a system in order to solve the system?
How do you know your solution doesn’t contain any errors?
How does solving systems of equations compare to solving equations with variables on both sides like we did in earlier lessons?
When you solved equations with variables on both sides, some had one solution, some had no solutions, and some had infinite solutions. Do you think systems of equations can have no solutions or infinite solutions?

Activity Synthesis

The goal of this discussion is to deliberately connect the current topic of systems of equations to the previous topic of solving equations with variables on both sides. For each of the following questions, give students 30 seconds of quiet think time and then invite students per question to explain their answer. The final question looks ahead to the following activity.

“Do you need to see the graphs of the equations in a system in order to solve the system?" (No, but the graphs made me feel more confident that my answer was correct.)
“How do you know your solution doesn’t contain any errors?" (I know my solution does not have errors because I substituted my values for x and y into the equations and they made both equations true.)
“How does solving systems of equations compare to solving equations with variables on both sides like we did in earlier lessons?" (They are very similar, only with a system of equations you are finding an x and a y to make both equations true and not just an x to make one equation true.)
“When you solved equations with variables on both sides, some had one solution, some had no solutions, and some had infinite solutions. Do you think systems of equations can have no solutions or infinite solutions?" (Yes. We have seen some graphs of parallel lines where there were no solutions and some graphs of lines that are on top of one another where there are infinite solutions.)

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Different Types of Systems

Activity 2

MLR1: Stronger & Clearer Each Time

Launch

Remind students of the activity they did sorting equations with a single variable where each equation had either one solution, no solution, or infinitely many solutions. Tell them that, just like one variable equations, systems of equations can also have either one solution, no solution, or infinitely many solutions. Point out that in the previous activity, each of the three systems of equations had one solution, which they found algebraically by solving the system, and so the graphs of the equations of the system showed one point where the lines intersected. Ask students what they think the graphs of equations from systems with no or infinitely many solutions might look like. Allow 30 seconds of quiet think time before inviting a few students to suggest possibilities for each type of system while recording and displaying their ideas for all to see. Remind students of the activities in previous lessons where they have seen these situations and their graphs (a bike race between Elena and Jada had infinite solutions and stacking different sized cups had none).

Arrange students in groups of 2–3. Provide each group with access to straightedges and scissors as well as one copy of the blackline master. Encourage partners to split the work by cutting apart the problems, each taking one to three graphs, and then trading pages within their group to check the work. Give 4–6 minutes for groups to complete the graphs and remind students to use straightedges for precision while graphing.

Before beginning the final problem, have each group trade their work with another group and place a question mark next to the graphs they are not sure are correct. Give groups 3–4 minutes to revise as needed and write their descriptions for the second problem followed by a whole-class discussion.

If using the digital activity, use the discussion structure above. The digital applet will make the graphing and solving of systems go quickly so students can spend more time analyzing the solutions. Using technology to graph allows students to focus on the main purpose of the lesson and also recognize the value in technology when solving systems in addition to appreciating when the graphing method is efficient. In this activity, one of the main purposes is to notice what is common among systems with the same number of solutions. Therefore, it may be useful to ask students to justify why the lines graphed with no obvious intersections are actually parallel.

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What do you think the graphs of equations from systems with infinitely many or no solutions might look like?

Take 4-6 minutes to work on this part and then pause for instructions before moving on.

Cut apart the problems
Each of you take 1-3 problems to graph
Trade pages with your group to check work

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Let’s Revise

Trade back

Trade papers with the group next to you.
Place a question mark next to any graphs you are not sure are correct.

Revise your work if needed
Write your description for #2.

2. Describe what the graph of a system of equations looks like when it has . . .

1 solution 0 solutions infinitely many solutions

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Let’s Talk About It

I am going to assign your group a type of system (1, 0, or infinitely many solutions)
Write a system of equations that would have that number of solutions

Activity Synthesis

The goal of this discussion is for students to draw conclusions about the relationship between the number of solutions a system of equations has and the appearance of the graphs of the equations in the system. Select 2–3 students to share and explain their answers to the second problem.

If no students mention it, bring in slope language and how inspecting the slopes of the equations before graphing or solving can give clues to the possible number of solutions the system has. In particular, students should notice that systems with lines that have different slopes have a single solution, lines that have the same slope and different y-intercept have no solution, and lines that have the same slope and y-intercept will have infinitely many solutions.

Assign a number of solutions (one, none, or infinite) to each group and ask them to write a system of equations that would have that number of solutions. Have a few groups share their systems and describe how the graphs of the systems would look. In particular, ask each group to describe how the slope and y-intercept of their written lines would be seen in the graph and how the number of solutions would appear on the graph. Following the description, display the graph of the system using a digital resource, if possible, or a general sketch on a set of displayed axes.

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Are you ready for more?

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Lesson Synthesis

How can you know the number of solutions for a system of equations from its graph?
How can you know the number of solutions for a system of equations from their equations?

To highlight the connection between the number of solutions to a system of equations and features of its graph and equations, ask:

“How can you know the number of solutions for a system of equations from its graph?” (If the two lines intersect at a point, there is one solution. If the two lines are parallel and do not intersect, there are no solutions. If the two lines are drawn through the same points, there are infinitely many solutions.)
“How can you know the number of solutions for a system of equations from their equations?” (If the two equations have different slopes, there is one solution. If the two equations have the same slope and different y-intercepts, there are no solutions. If the two equations have the same slope and the same y-intercept, there are infinitely many solutions.)

If students do not make the connection themselves, remind them of their earlier conclusions about the number of solutions and equation in one variable has.

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Today’s Goals

I can graph a system of equations.
I can solve systems of equations using algebra.

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Two Lines

Cool Down