Lesson 13

This Math Talk encourages students to look for connections between the features of graphs and of linear equations that each represent a system. Given two graphs on an unlabeled coordinate plane, students must rely on what they know about horizontal and vertical lines, intercepts, and slope to determine if the graphs could represent each pair of equations. The equations presented and the reasoning elicited here will be helpful later in the lesson, when students solve systems of equations by substitution.

All four systems include an equation for either a horizontal or a vertical line. Some students may remember that the equation for such lines can be written as \(x = a\) or \(y=b\), where \(a\) and \(b\) are constants. (In each of the first three systems, one equation is already in this form. In the last system, a simple rearrangement to one equation would put it in this form.) Activating this knowledge would enable students to quickly tell whether a system matches the given graphs.

Those who don't recall it can still reason about the system structurally. For instance, given a system with \(x=\text-5\) as one of the equations, they may reason that any point that has a negative \(x\)-value will be to the left of the vertical axis. The solution (if there is one) to this system would have to have -5 for the \(x\)-value. The intersection of the given graphs is a point to the right of the vertical axis (and therefore having a positive \(x\)-value), so the graphs cannot represent that system.

To match graphs and equations, students need to look for and make use of structure (MP7) in both representations. In explaining their strategies, students need to be precise in their word choice and use of language (MP6). 

Because the warm-up is intended to promote reasoning, discourage the use of graphing technology to graph the systems.

Display one system at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

If no students mentioned solving the systems and then checking to see if the solution could match the graphs, ask if anyone approached it that way. For instance, ask: “How could we find the solution to the second system without graphing?” Give students a moment to discuss their ideas with a partner and then proceed to the next activity.

In this activity, students see the same four pairs of equations as those in the warm-up. This time, their job is to find a way to solve the systems. Some students may choose to solve by graphing, but the systems lend themselves to be solved efficiently and precisely by substitution.

As students work, pay attention to the methods students use to solve the systems. Identify those who solve by substitution—by replacing a variable or an expression in one equation with an equal value or equivalent expression from the other equation. Ask these students to share later.

Arrange students in groups of 2. Give students 6–8 minutes of quiet time to solve as many systems as they can and then a couple of minutes to share their responses and strategies with their partner.

Some students may not remember to find the value of the second variable after finding the first. They may need a reminder that the solution to a system of linear equations is a pair of values.

If some students struggle with the last system because the variable that is already isolated is equal to an expression rather than a number, ask what they would do if the first equation were \(y= \text{a number}\) instead of \(y=2x-7\).

If students don't know how to approach the last system, ask them to analyze both equations and see if the value of one of the variables could be found easily.

Select previously identified students to share their responses and strategies. Display their work for all to see. Highlight the strategies that involve substitution and name them as such.

Make sure students see that the last two equations can be solved by substituting in different ways. Here are two ways for solving the third system, \(\begin{cases} 3x = 8\\3x + y = 15 \end{cases} \), by substitution:

Here are two ways of solving the last system, \(\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}\), by substitution:

In each of these two systems, students are likely to notice that one way of substituting is much quicker than the other. Emphasize that when one of the variables is already isolated or can be easily isolated, substituting the value of that variable (or the expression that is equal to that variable) into the other equation in the system can be an efficient way to solve the system.

The activity allows students to practice solving systems of linear equations by substitution and reinforces the idea that there are multiple ways to perform substitution. Students are directed to find the solutions without graphing.

Monitor for the different ways that students use substitutions to solve the systems. Invite students with different approaches to share later.

Keep students in groups of 2. Give students a few minutes to work quietly and then time to discuss their work with a partner. If time is limited, ask each partner to choose two different systems to solve. Follow with a whole-class discussion.

When solving the second system, students are likely to substitute the expression \(2m+10\) for \(p\) in the first equation, \(2m-2p=\text-6\). Done correctly, it should be written as \(2m-2(2m+10)=\text-6\). Some students may neglect to write parentheses and write \(2m-4m+10=\text-6\). Remind students that if \(p\) is equal to \(2m+10\), then \(2p\) is 2 times \(2m+10\) or \(2(2m+10)\). (Alternatively, use an example with a sum of two numbers for \(p\): Suppose \(p=10\), which means \(2p=2(10)\) or 20. If we express \(p\) as a sum of 3 and 7, or \(p=3+7\), then \(2p=2(3+7)\), not \(2\boldcdot 3 + 7\). The latter has a value of 13, not 20.)

Some students who correctly write \(2m-2(2m+10)=\text-6\) may fail to distribute the subtraction and write the left side as \(2m-4m+20\). Remind them that subtracting by \(2(2m+10)\) can be thought of as adding \(\text-2(2m+10)\) and ask how they would expand this expression.

Select previously identified students to share their responses and reasoning. Display their work for all to see.

Highlight the different ways to perform substitutions to solve the same system. For example:

• In the second system, \(\begin{cases} 2m – 2p = \text-6\\ p = 2m + 10 \end{cases}\), we could substitute \(2m+10\) for \(p\) in the first equation, or we could substitute \(p-10\) for \(2m\) in the first equation.
• In the third system, \(\begin{cases} 2d = 8f \\ 18 - 4f = 2d \end{cases}\), we could substitute \(8f\) for \(2d\) in the second equation, or we could substitute \(\frac14 d\) for \(f\) in the second equation.