Lesson 15

Writing Systems of Equations

15.1: How Many Solutions? Matching (5 minutes)

Warm-up

This warm-up asks students to connect the algebraic representations of systems of equations to the number of solutions. Efficient students will recognize that this can be done without solving the system, but rather using slope, \(y\)-intercept, or other methods for recognizing the number of solutions.

Monitor for students who use these methods:

  1. Solve the systems to find the number of solutions.
  2. Use the slope and \(y\)-intercept to determine the number of solutions.
  3. Manipulate the equations into another form, then compare the equations.
  4. Notice that the left side of the second equation in system C is double the left side of the first equation, but the right side is not.

Launch

Arrange students in groups of 2. Tell students that each number can be used more than once. Allow students 2 minutes of work time followed by a whole-class discussion.

Student Facing

Match each system of equations with the number of solutions the system has.

  1. \(\begin{cases} y=\text-\frac43x+4 \\ y = \text-\frac43x-1 \end{cases}\)
  2. \(\begin{cases} y=4x-5 \\ y = \text-2x+7 \end{cases}\)
  3. \(\begin{cases} 2x+3y = 8 \\ 4x+6y = 17 \end{cases}\)
  4. \(\begin{cases} y= 5x-15 \\ y= 5(x-3) \end{cases}\)
  1. No solutions
  2. One solution
  3. Infinitely many solutions

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is to bring out any methods students used to find the number of solutions for the systems.

Select previously identified students to share their methods for finding the number of solutions in the sequence given in the narrative. After each student shares their method, ask the class which method they preferred to answer the given question. Connect each problem to the concepts learned in the previous lesson by asking students to describe how the graphs of the lines for each system might intersect.

15.2: Situations and Systems (10 minutes)

Activity

In this activity, students are presented with a number of scenarios that could be solved using a system of equations. Students are not asked to solve the systems of equations, since the focus at this time is for students to understand how to set up the equations for the system and to understand what the solution represents in context. 

Launch

Arrange students in groups of 2. Suggest that groups split up the problems so that one person works on the first and third problem while their partner works on the second and fourth. Students may work with their partners to get help when they are stuck, but are encouraged to try to set up the equations on their own first. Partners should discuss their systems and interpretation of the solution after each has had a chance to work on their own.

Allow students 5–7 minutes of partner work time followed by a whole-class discussion.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2–3 of the situations to complete.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

For each situation:

  • Create a system of equations.
  • Then, without solving, interpret what the solution to the system would tell you about the situation.
  1. Lin’s family is out for a bike ride when her dad stops to take a picture of the scenery. He tells the rest of the family to keep going and that he’ll catch up. Lin's dad spends 5 minutes taking the photo and then rides at 0.24 miles per minute until he meets up with the rest of the family further along the bike path. Lin and the rest were riding at 0.18 miles per minute.
  2. Noah is planning a kayaking trip. Kayak Rental A charges a base fee of \$15 plus \$4.50 per hour. Kayak Rental B charges a base fee of \$12.50 plus \$5 per hour.
  3. Diego is making a large batch of pastries. The recipe calls for 3 strawberries for every apple. Diego used 52 fruits all together.
  4. Flour costs \$0.80 per pound and sugar costs \$0.50 per pound. An order of flour and sugar weighs 15 pounds and costs \$9.00.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The focus of the discussion should be on making sense of the context and interpreting the solutions within the context of the problems.

Invite groups to share their systems of equations and interpretation of the solution for each problem. As groups share, record their systems of equations for all to see. When necessary, ask students to explain the meaning of the variables they used. For example, \(t\) represents the number of minutes the family rides after Lin’s dad starts riding again after taking the picture.

To highlight the connections between the situations and the equations that represent them, ask:

  • “How many solutions will each of these systems of equations have?” (Each system has exactly one solution. I can tell this because the slopes of each pair of equations are different.)
  • “If Lin’s dad biked 0.17 miles per minute instead of 0.24 miles per minute, how would that change the system of equations?” (The first equation would be \(d = 0.17t\).)
    • “How many solutions would there be for this new system where Lin’s dad rides slower?” (Based on the equations there should still be one solution.)
    • “Would Lin’s dad ever catch up with the family?” (He would not. He started farther back and rides slower than the family. The solution to the system would have a negative value for time which does not make sense in the context of the problem.)

If students disagree that there is a solution to the modified first problem in which Lin’s dad rides slower than the family, you may display the graph of the modified system and point out the point where the lines intersect. So, although the system has a solution, it is disregarded in this context since it does not make sense.

Representing, Conversing: MLR3 Clarify, Critique, Correct. To help students make sense of the solution to a system of equations, offer an incorrect or ambiguous response for the problem about Noah’s kayaking trip. After revealing the correct system of equations, display this ambiguous statement: “This solution represents the same amount of money for both rental places.” Ask pairs of students to clarify the meaning of this response and then critique it. Invite pairs to offer a correct response by asking “What language might you add or change to make this statement more accurate?” Improved responses should include a reference to both variables (e.g. the amount of time and cost). This will help students interpret the solution to a system of equations in a specific context.
Design Principle(s): Maximize meta-awareness; Support sense-making

15.3: Info Gap: Racing and Play Tickets (20 minutes)

Activity

In this activity students have an opportunity to apply what they know about systems of linear equations to solve a problem about a real-world situation. One equation for each situation is given. Students may choose to write another equation to create a system that represents the constraints in the problem, and then solve the system algebraically or by graphing. Another possible strategy would be to pull quantities out of the given equation and solve the problem arithmetically. Monitor for students who use each of these strategies to share during the whole-class discussion.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Two problem cards. 2 data cards. 

Launch

Arrange students in groups of 2. Provide access to geometry toolkits. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for the second problem, in which students switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve a problem about a real-world situation by applying what they know about systems of linear equations. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select students with different strategies to share their approaches to each question, starting with less efficient methods and ending with more efficient methods.

15.4: Solving Systems Practice (10 minutes)

Optional activity

In this activity, students solve a variety of systems of equations, some involving fractions, some involving substitution, and some involving inspection. This gives students a chance to practice using the methods they have learned in this section for solving systems of equations to solidify that learning. Some of the systems listed are ones students could have used in an earlier activity in this lesson, to describe the situations there. In the discussion, students compare the systems here to the ones they wrote in that activity and interpret the answer in that context.

Launch

Keep students in groups of 2. Allow students 5–7 minutes of partner work time followed by a whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

Here are a lot of systems of equations:

  • \(\begin{cases} y=\text-2x+6 \\ y=x-3 \end{cases}\)
  • \(\begin{cases} y=5x-4 \\ y=4x+12 \end{cases}\)
  • \(\begin{cases} y=\frac23x-4 \\ y=\text-\frac43x+9 \end{cases}\)
  • \(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)
  • \(\begin{cases} y=x - 6\\ x=6 + y \end{cases}\)
  • \(\begin{cases} y=0.24x\\ y=0.18x+0.9 \end{cases}\)
  • \(\begin{cases} y=4.5x+15 \\ y=5x+12.5 \end{cases}\)
  • \(\begin{cases} y=3x \\ x+y=52 \end{cases}\)
  1. Without solving, identify 3 systems that you think would be the least difficult for you to solve and 3 systems you think would be the most difficult. Be prepared to explain your reasoning.
  2. Choose 4 systems to solve. At least one should be from your "least difficult" list and one should be from your "most difficult" list.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

There are two key takeaways from this discussion. The first is to reinforce that some systems can be solved by reasoning whether its possible for a solution to one equation to also be a solution to another. The second takeaway is that there are some systems that students will only be able to solve after learning techniques in future grades.

For each problem, ask students to indicate if they identified the system as least or most difficult. Record the responses for all to see.

Bring students' attention to this system:

\(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)

Ask students what the two equations in the system have in common to each other and to think about whether a solution to the first equation could also be a solution to the second. One can reason there is no solution because \(4y+7x\) cannot be equal to both 6 and -5.

Ask students to return to the earlier activity and see if they can find any of those systems in these problems. (Lin's family ride is the sixth system. Noah's kayaking trip is the seventh system. Diego's baking is the eighth system.) After students notice the connection, invite students who chose those systems to solve to interpret the numerical solutions in the contexts from the earlier activity.

Students may be tempted to develop the false impression that all systems where both equations are given as linear combinations can be solved by inspection. Conclude the discussion by displaying this system that students defined in the last activity about sugar and flour:

\(\begin{cases} 0.8x+0.5y = 9 \\ x+y = 15 \end{cases}\)

Tell students that this system has one solution and they will learn more sophisticated techniques for solving systems of equations like this in future grades.

Speaking, Listening, Conversing: MLR8 Discussion Supports. Before the whole-class discussion, use this routine to give students an opportunity to discuss their reasons for labeling systems of equations as “least difficult” or “most difficult.” Provide the following sentence frame: “_____ is the least/most difficult to solve because _____.” Encourage listeners to press for detail by asking questions such as “What would make that problem easier to solve?”; “What do your least/most difficult problems have in common?” This will help students make connections between the structures of given systems of equations, and possible strategies they can use to solve them.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

To wrap up the lessons on solving systems of equations, consider displaying the three systems of equations and asking students how they might begin to solve the systems.

\(\begin{cases} y = 2x + 1 \\ y = \frac{1}{2}x + 10 \end{cases}\) (Both graphing and substitution methods work well)

\(\begin{cases} x = 5- 2y \\ 2x + 6y = 16 \end{cases}\) (Substitution works best)

\(\begin{cases} 5x + 4y = 20 \\ 10x + 8y = 60 \end{cases}\) (Inspection may work best)

If there is time, consider assigning each system to small groups for them to solve, then share their solutions with the class.

15.5: Cool-down - Solve This (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We have learned how to solve many kinds of systems of equations using algebra that would be difficult to solve by graphing. For example, look at

\(\begin{cases} y = 2x -3 \\ x+2y=7 \end{cases}\)

The first equation says that \(y=2x-3\), so wherever we see \(y\), we can substitute the expression \(2x-3\) instead. So the second equation becomes \(x+2(2x-3) = 7\).

We can solve for \(x\):

\(\begin{align} x+4x-6 &= 7 &&\text{distributive property}\\ 5x-6 &=7 &&\text{combine like terms}\\ 5x &= 13 &&\text{add 6 to each side}\\ x&= \frac{13}{5} && \text{multiply each side by} \frac{1}{5} \end{align}\)

We know that the \(y\) value for the solution is the same for either equation, so we can use either equation to solve for it. Using the first equation, we get:

\(\begin{align} y &= 2(\frac{13}{5})-3 &&\text{substitute \(x= \frac{13}{5}\) into the equation}\\ y &=\frac{26}{5}-3 &&\text{multiply \(2(\frac{13}{5})\) to make \(\frac{26}{5}\)} \\ y &=\frac{26}{5}-\frac{15}{5} &&\text{rewrite 3 as \(\frac{15}{5}\)}\\ &y=\frac{11}{5} \end{align}\)

If we substitute \(x=\frac{13}5\) into the other equation, \(x+2y=7\), we get the same \(y\) value. So the solution to the system is \(\left(\frac{13}{5},\frac{11}5\right)\).

There are many kinds of systems of equations that we will learn how to solve in future grades, like \(\begin{cases} 2x+3y = 6 \\ \text-x+2y = 3 \end{cases}\).

Or even \(\begin{cases} y = x^2 +1 \\ y = 2x+3 \end{cases}\).