# Lesson 13

Solving Systems by Substitution

## 13.1: Math Talk: Is It a Match? (10 minutes)

### Warm-up

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.

### Launch

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.

*Representation: Internalize Comprehension.*To support working memory, provide students with sticky notes or mini whiteboards.

*Supports accessibility for: Memory; Organization*

### Student Facing

Here are graphs of two equations in a system.

Determine if each of these systems could be represented by the graphs. Be prepared to explain how you know.

\(\begin{cases} x + 2y = 8 \\x = \text-5 \end{cases}\)

\(\begin{cases} y = \text-7x + 13 \\y = \text-1 \end{cases}\)

\(\begin{cases} 3x = 8\\3x + y = 15 \end{cases}\)

\(\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}\)

### Student Response

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### Activity Synthesis

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.

*Speaking: MLR8 Discussion Supports.*Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because….” or “I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

*Design Principle(s): Optimize output (for explanation)*

## 13.2: Four Systems (15 minutes)

### 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.

### Launch

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.

*Engagement: Internalize Self Regulation.*Chunk this task in half to differentiate the degree of difficulty or complexity. Present and discuss the first two problems before showing the last two. Support students in metacognitive development by having students articulate the steps they used to solve by substitution in these more straightforward examples. Record the steps on a chart or whiteboard to encode the process for later retrieval. Then, reveal the remaining problems.

*Supports accessibility for: Organization; Attention*

### Student Facing

Here are four systems of equations you saw earlier. Solve each system. Then, check your solutions by substituting them into the original equations to see if the equations are true.

A\(\begin{cases} x + 2y = 8 \\x = \text-5 \end{cases}\)

B\(\begin{cases} y = \text-7x + 13 \\y = \text-1 \end{cases}\)

C\(\begin{cases} 3x = 8\\3x + y = 15 \end{cases}\)

D\(\begin{cases} y = 2x - 7\\4 + y = 12 \end{cases}\)

### Student Response

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### Anticipated Misconceptions

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.

### Activity Synthesis

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:

Finding the value of \(x\) and substituting it

into \(3x+8=15\):

\(\begin {align} 3x&=8\\x&=\frac83\\ \\3x+y &=15\\ 3(\frac83) + y &=15\\8+y &=15\\y&=7 \end{align}\)

Substituting the value of \(3x\) into \(3x+8=15\):

\(\begin {align} 3x+y &=15\\ 8 + y &=15\\y&=7 \end{align}\)

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

Substituting \(2x - 7\) for \(y\) in the equation \(4 + y = 12\):

\(\begin {align} 4+y&=12\\4 + (2x-7) &=12\\4 + 2x - 7 &=12\\ 2x -7 + 4 &=12\\ 2x-3&=12\\2x &=15\\x &=7.5\\ \\y&=2x - 7\\y&=2(7.5) - 7\\ y&=15-7\\y&=8 \end{align}\)

Rearranging or solving \(4+ y=12\) to get \(y = 8\), and then substituting 8 for \(y\) in the equation \(y=2x - 7\):

\(\begin {align} y&=2x - 7\\8&=2x - 7\\ 15&=2x \\ 7.5 &=x \end{align}\)

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.

## 13.3: What about Now? (10 minutes)

### Activity

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.

### Launch

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.

*Representation: Internalize Comprehension.*Differentiate the degree of difficulty or complexity by allowing students to complete problems in any order. Students may benefit from solving the system that uses the same letter variables \(( x, y)\) first. Highlight connections between representations by inviting students use the same color to highlight a variable and the expression to be substituted for that variable.

*Supports accessibility for: Conceptual processing*

### Student Facing

Solve each system without graphing.

\(\begin{cases} 5x – 2y = 26 \\ y + 4 = x \end{cases}\)

\(\begin{cases} 2m – 2p = \text-6\\ p = 2m + 10 \end{cases}\)

\(\begin{cases} 2d = 8f \\ 18 - 4f = 2d \end{cases}\)

\(\begin{cases} w + \frac17z = 4 \\ z = 3w –2 \end{cases}\)

### Student Response

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### Student Facing

#### Are you ready for more?

Solve this system with four equations.\(\begin{cases}3 x + 2y - z + 5w= 20 \\ y = 2z-3w\\ z=w+1 \\ 2w=8 \end{cases}\)

### Student Response

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### Anticipated Misconceptions

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.

### Activity Synthesis

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.

*Representing, Conversing: MLR7 Compare and Connect.*As students share their responses and reasoning with the class, call students’ attention to the different ways their classmates have chosen to make substitutions. Wherever possible, amplify student words and actions that involve the language of substitution: replace, substitute, solve for, and isolate.

*Design Principle(s): Maximize meta-awareness; Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

To emphasize that the method we choose for solving a systems may depend on the system, and that some systems are more conducive to be solved by substitution than others, present the following systems to students:

System 1

\(\begin {cases} 3m + n = 71\\2m-n =30 \end {cases}\)

System 2

\(\begin {cases} 4x + y = 1\\y = \text-2x+9 \end {cases}\)

System 3

\(\displaystyle \begin{cases} 5x+4y=15 \\ 5x+11y=22 \end{cases}\)

Ask students to choose a system and make a case (in writing, if possible) for why they would or would not choose to solve that system by substitution. Consider asking students to use sentence starters such as these:

- I would choose to solve system \(\underline{\hspace{0.5in}}\) by substitution because . . .
- I would
*not*choose to solve system \(\underline{\hspace{0.5in}}\) by substitution because . . .

With a little bit of rearrangement, all systems could be solved by substitution without cumbersome computation, but system 2 would be most conducive to solving by substitution.

Consider collecting students' responses or asking them to share their written arguments with a partner.

## 13.4: Cool-down - A System to Solve (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

The solution to a system can usually be found by graphing, but graphing may not always be the most precise or the most efficient way to solve a system.

Here is a system of equations:

\(\begin {cases} 3p + q = 71\\2p - q = 30 \end {cases}\)

The graphs of the equations show an intersection at approximately 20 for \(p\) and approximately 10 for \(q\).

Without technology, however, it is not easy to tell what the exact values are.

Instead of solving by graphing, we can solve the system algebraically. Here is one way.

If we subtract \(3p\) from each side of the first equation, \(3p + q = 71\), we get an equivalent equation: \(q= 71 - 3p\). Rewriting the original equation this way allows us to isolate the variable \(q\).

Because \(q\) is equal to \(71-3p\), we can substitute the expression \(71-3p\) in the place of \(q\) in the second equation. Doing this gives us an equation with only one variable, \(p\), and makes it possible to find \(p\).

\(\begin {align} 2p - q &= 30 &\quad& \text {original equation} \\ 2p - (71 - 3p) &=30 &\quad& \text {substitute }71-3p \text{ for }q\\ 2p - 71 + 3p &=30 &\quad& \text {apply distributive property}\\ 5p - 71 &= 30 &\quad& \text {combine like terms}\\ 5p &= 101 &\quad& \text {add 71 to both sides}\\ p &= \dfrac{101}{5} &\quad& \text {divide both sides by 5} \\ p&=20.2 \end {align}\)

Now that we know the value of \(p\), we can find the value of \(q\) by substituting 20.2 for \(p\) in either of the original equations and solving the equation.

\(\begin {align} 3(20.2) + q &=71\\60.6 + q &= 71\\ q &= 71 - 60.6\\ q &=10.4 \end{align}\)

\(\begin {align} 2(20.2) - q &= 30\\ 40.4 - q &=30\\ \text-q &= 30 - 40.4\\ \text-q &= \text-10.4 \\ q &= \dfrac {\text-10.4}{\text-1} \\ q &=10.4 \end {align}\)

The solution to the system is the pair \(p=20.2\) and \(q=10.4\), or the point \((20.2, 10.4)\) on the graph.

This method of solving a system of equations is called solving by **substitution**, because we substituted an expression for \(q\) into the second equation.