Lesson 15

The purpose of this warm-up is to encourage students to use substitution to solve equations mentally and see that sometimes they cannot use this method.

Display each problem one at a time. Give students 30 seconds of quiet think time followed by a whole-class discussion. Leave each problem displayed throughout the talk. 

After each problem, ask students to share their solutions. Record and display their responses for all to see. As students share their strategies, highlight the term substitution as a strategy to solve an equation. For the third question, ask students why they could not use the same strategy as they did in the earlier questions.

In this activity, students solve systems of linear equations that lend themselves to substitution. There are 4 kinds of systems presented: one kind has both equations given with the \(y\) value isolated on one side of the equation, another kind has one of the variables given as a constant, a third kind has one variable given as a multiple of the other, and the last kind has one equation given as a linear combination. This progression of systems nudges students towards the idea of substituting an expression in place of the variable it is equal to.

Notice which kinds of systems students think are least and most difficult to solve.

In future grades, students will manipulate equations to isolate one of the variables in a linear system of equations. For now, students do not need to solve a system like \(x+2y=7\) and \(2x-2y=2\) using this substitution method.

Arrange students in groups of 2. Give students 10 minutes of quiet work time. Encourage students to check in with their partner between questions. Tell students that if there is disagreement, they should work to reach an agreement. Follow with a whole-class discussion.

Some students may have trouble transitioning from systems where both equations are given with one variable isolated to other kinds of systems. Ask these students to look at a system where one of the variables is given as a constant. For example, ask them to look at equation B:

\(\displaystyle \begin{cases} y= 7 \\ x=3y-4 \end{cases}\)

Ask, “If \(y\) is equal to 7, then what is \(3y\) equal to?” If a student continues to struggle, refer them back to this example and then ask, “In this new problem, do we know what expression \(y\) (or \(x\)) is equal to? Then whenever we see \(y\) (or \(x\)), what can we replace it with instead?”

This discussion has a two main takeaways. The first is to formalize the idea of substitution in a system of equations. Another is to recognize systems where both equations are written with one variable isolated are actually special cases of substitution.

Invite students to share which systems they thought would be easiest to solve and which would be hardest. To involve more students in the conversation, consider asking:

• “Did you change your mind about any of the systems being more or less difficult after you solved them?”
• “What was similar in these problems? What was different?” (The systems vary slightly in how they are presented, but all of the problems can be solved by replacing a variable with an expression it is equal to.)
• “Will your strategy work for the other systems in this list?” (Yes, substitution works in all the given problems.)

Tell students that the key underlying concept for all of these problems is that it is often helpful to replace a variable with the expression it is equal to, and that this “replacing” is called “substitution.” Point out that even setting the expressions for \(y\) in the first two problems equal to each other is really substituting \(y\) in one equation with the expression it is equal to as given by the other equation. It may be helpful for students to hear language like, “Since \(y\) is equal to \(\text-2x\), that means wherever I see \(y\), I can substitute in \(\text-2x\).”

In this activity, students are asked to make sense of Tyler’s justification for the number of solutions to the system of equations (MP3). This activity continues the thread of reasoning about the structure of an equation (MP7) and the focus should be on what, specifically, in the equations students think Tyler sees that makes him believe the system has no solutions.

Give students 2–3 minutes of quiet think time to read the problem and decide if they agree or disagree with Tyler. Use the remaining time for a whole-class discussion.

The goal of this discussion is to look at one way to reason about the structure of a system of equations in order to determine the solution and then have students make their own reasoning about a different, but similar, system of equations.

Poll the class to see how many students agree with Tyler and how many students disagree with Tyler. If possible, invite students from each side to explain their reasoning. As students explain, it should come out that Tyler is correct and, if no student brings up the idea, make sure to point out that we can also visualize this by graphing the equations in the system and noting that the lines look parallel and will never cross.

In the previous activity, students noticed that if they knew what one variable was equal to, they could substitute that value or expression into another equation in the same problem. Point out that, in this problem, the expression \((x + y)\) is equal to 5 in the first equation. If, in the second equation, we replace \((x+y)\) with 5, the resulting equation is \(5 = 7\) which cannot be true regardless of the choice of \(x\) and \(y\).

Display the following system and ask students how many solutions they think it has and to give a signal when they think they know:

\(\displaystyle \begin{cases} 4x + 2y = 8\\2x + y = 5 \end{cases}\)

Once the majority of the class signals they have an answer, invite several students to explain their thinking. There are multiple ways students might use to reason about the number of solutions this system has. During the discussion, encourage students to use the terms coefficient and constant term in their reasoning. Introduce these terms if needed to help students recall their meanings. Bring up these possibilities if no students do so in their explanations:

• “Re-write the second equation to isolate the \(y\) variable and substitute the new expression into the first equation in order to find that the system of equations has no solutions.”
• “Notice that both equations are lines with the same slope but different \(y\)-intercepts, which means the system of equations has no solutions.”
• “Notice that \(4x+2y\) is double \(2x+y\), but 8 is not 5 doubled, so the system of equations must have no solution.”