Lesson 16

In this warm-up, experiment with the graphical effects of multiplying both sides of an equation in two variables by a factor.

Students are prompted to multiply one equation in a system by several factors to generate several equivalent equations. They then graph these equations on the same coordinate plane that shows the graphs of the original system. Students notice that no new graphs appear on the coordinate plane and reason about why this might be the case.

The work here reminds students that equations that are equivalent have all the same solutions, so their graphs are also identical. Later, students will rely on this insight to explain why we can multiply one equation in a system by a factor—which produces an equivalent equation—and solve a new system containing that equation instead.

Display the equation \(x+5 = 11\) for all to see. Ask students:

• "What is the solution to this equation?" (\(x=6\))
• "If we multiply both sides of the equation by a factor, say, 4, what equation would we have?" (\(4x + 20 = 44\)) "What is the solution to this equation?" (\(x=6\))
• "What if we multiply both sides by 100?" (\(100x + 500 = 1,\!100\)) "By 0.5?" (\(0.5x + 2.5 = 5.5\))
• "Is the solution to each of these equations still \(x=6\)?" (Yes)

Remind students that these equations are one-variable equations, and that multiplying both sides of a one-variable equation by the same factor produces an equivalent equation with the same solution. 

Ask students: "What if we multiply both sides of a two-variable equation by the same factor? Would the resulting equation have the same solutions as the original equation?" Tell students that they will now investigate this question by graphing. 

Arrange students in groups of 2–4 and provide access to graphing technology to each group. To save time, consider asking group members to divide up the tasks. (For example, one person could be in charge of graphing while the others write equivalent equations, and everyone analyze the graphs together.)

Invite students to share their observations about the graphs they created. Ask students why the graphs of equations A1, A2, and A3 all coincide with the graphs of the original equation A. Discuss with students:

• "How can we explain the identical graphs?" (Equations A1, A2, and A3 are equivalent to equation A, so they all have the same solutions as equation A, and their graphs are the same line as the graph of A.)
• "What move was made to generate A1, A2, and A3? Why did it create equations that are equivalent to A?" (The two sides of equation A was multiplied by the same factor, which keeps the two sides equal.)

To further illustrate that equations A1, A2, and A3 are equivalent to equation A, and if time permits, consider:

• Using tables to visualize the identical \((x,y)\) pairs. For example:

  \(4x+y=1\)

  x	y
  0	1
  1	-3
  2	-7
  3	-11

  \(20x+5y=5\)

  x	y
  0	1
  1	-3
  2	-7
  3	-11

  \(2x+\frac12y=\frac12\)

  x	y
  0	1
  1	-3
  2	-7
  3	-11

• Reminding students that, earlier in the unit, they saw that isolating one variable is a way to see if two equations are equivalent. If we isolate \(y\) in equations A, A1, A2, and A3, the rearranged equation will be identical: \(y=\text-4x+1\). 

In an earlier lesson, students learned that adding the two equations in a system creates a new equation that shares a solution with the system. In the warm-up, they saw that multiplying an equation by a factor creates an equivalent equation that shares all the same solutions as the original equation.

Here students learn that each time we perform a move that creates one or more new equations, we are in fact creating a new system that is equivalent to the original system. Equivalent systems are systems with the same solution set, and we can write a series of them to help us get closer to finding the solution of an original system.

For instance, if \(4x+y=1\) and \(x+2y=9\) form a system, and \(4x+8y=36\) is a multiple of the second equation, the equations \(4x+y=1\) and \(4x+8y=36\) form an equivalent system that can help us eliminate \(x\) and make progress toward finding the value of \(y\).

Students also learn to make an argument that explains why each new system is indeed equivalent to the one that came before it (MP3), building on the work of justifying equivalent equations in earlier lessons.

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, and then 1–2 minutes to discuss their thinking with their partner. Follow with a whole-class discussion.

Some students may be confused by the subtraction symbol after the second equation, wondering if some number is supposed to appear after the sign. Encourage them to ignore the sign at first, find the relationship between the three equations, and then think about what the sign might mean.

Invite students to share their analyses of Elena's moves. Highlight responses that point out that Elena's moves enabled her to eliminate the \(x\)-variable. (In other words, multiplying equation B by 4 gives \(4x + 8y=36\) and subtracting this equation from equation A removes the \(x\)-variable.)

Display for all to see the two original equations in the system and the new equations Elena wrote (\(4x + 8y=36\) and \(\text-7y=\text-35\)). Then, ask students to predict what the graphs of all four equations might look like.

Next, use graphing technology to display all four graphs. Invite students to share their observations about the graphs.

Students are likely to observe that the graphs all intersect at the same point, \((\text-1,5)\) and that there are only three lines, instead of 4. Discuss why only three lines are visible. Make sure students understand that this is because the equations \(x+2y =9\) (equation B) and \(4x+8y=36\) are equivalent, so they share all the same solutions.

Then, focus students' attention on two things: the series of systems that came into play in solving the original system, and the explanations that justify each step along the way. Display the following systems and sequence the discussion as follows:

Explain that what we have done was to create equivalent systems—systems with the exact same solution set—to help us get closer and closer to the solution of the original system.

One way to create an equivalent system is by multiplying one or both equations by a factor. It helps to choose the factor strategically—one that would allow one variable to be eliminated when the two equations in the new system are added or subtracted. Elena chose to multiply equation B by 4 so that the \(x\)-variable could be eliminated.

Ask students:

• "Suppose we want to solve the system by first eliminating the \(y\)-variable. What factor should we choose? Which equation should we apply it to?" (We could multiply equation A by 2, or multiply equation B by \(\frac12\).)
• "Could we multiply equation A by 6 and multiply equation B by -3 as a way to eliminate the \(y\)-variable?" (Yes. Each new equation (\(24x+6y=6\) and \(\text-3x -6y = \text-27\)) would be equivalent to the original. Adding the two new equations would eliminate the \(y\)-variable.)

Earlier, students encountered the idea of equivalent systems. They described the moves that lead to equivalent equations and explained why the solution to those equations was the same as the solution to the original system. This activity reinforces that work.

Given an unordered set of equivalent systems, students work with a partner to arrange the systems such that the order constitutes logical steps toward solving an original system. Along the way, they take turns describing how each system came from the previous one and explaining how they know it has the same solution as its predecessor. The work allows students to practice constructing logical arguments and critiquing those of others (MP3).

Here is an image of the cards for reference and planning.

Arrange students in groups of 2. Give one set of pre-cut slips or cards from the blackline master to each group. Ask students to find the slip or card that shows a system of equations and is labeled “Start here.” 

Explain that all the other slips contain equivalent systems that represent steps in solving the starting system. Ask students to arrange the slips in the order that would lead to its solution, and to make sure they can explain what moves take each system to the next system and why each system is equivalent to the one before it.

Partners should take turns finding the next step in the solving process and explaining their reasoning. As one student explains, the partner's job is to listen and make sure they agree and the explanation makes sense. If they disagree, the partners should discuss until they reach an agreement.

Give students 5 minutes to arrange the systems. Follow with a whole-class discussion.

Students who are thinking algorithmically about solving systems of equations may think that the first step should involve multiplying each side of the second equation by \(\frac45\), becoming frustrated when no cards show that step. Encourage them to compare both the first and the second equations in the starting card to the first and second equations on other cards to gain some ideas about what steps might have been taken.

Ask students to share the order in which the systems should be arranged to lead to the solution. Display the ordered systems for all to see.

Point out to students that this particular solution path involves multiplying each of the two equations by a factor in order to eliminate the \(x\)-variable. Ask students if it's possible to eliminate a variable by multiplying only one equation by a factor. (Yes, we could multiply the first equation by \(\frac54\) to eliminate \(x\), or by 3 to eliminate \(y\). Or we could multiply the second equation by \(\frac45\) to eliminate \(x\), or by \(\frac13\) to eliminate \(y\).)

See Lesson Synthesis for discussion questions and ways to help students connect the ideas in the lesson.

This optional activity gives students another opportunity to identify moves that lead to equivalent systems and to explain why the resulting systems have the same solution (MP3). It also prompts students to build their own equivalent systems, which encourages them to think strategically about what to do to reach the goal of solving the system, rather than on a particular solution path.

As students work, prompt students to articulate the reasoning and assumptions. Ask questions such as:

• "How do you know the factor to use when multiplying an equation so a variable can be eliminated?"
• "Can the factor be a fraction? A negative number? Why or why not?"
• "Why doesn't adding one equation to another equation change the solution of the system?"

Keep students in groups of 2, if desired.

If students struggle to create an equivalent system of their own, ask them to start by deciding on a variable they'd like to eliminate. Then, ask them to think about a factor that, when multiplied to one equation, would produce the same or opposite coefficients for that variable. If they are uncomfortable using a fractional factor, ask if they could find a factor to apply to each equation such that the resulting equations have the same or opposite coefficients for the variable they wish to eliminate.

Invite students with different first steps to display their equivalent systems and solution paths. Prompt them (or others students) to explain why each system generated share the same solution as the original system or the system before it.

Verify that, regardless of the moves made, the different paths all led to the same pair of values.