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

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.

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 Equation A.)
• "What move was made to generate Equations A1, A2, and A3? Why did it create equations that are equivalent to Equation A?" (The two sides of Equation A were 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 pairs. For example:

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

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

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

• Remind 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 in Equations A, A1, A2, and A3, the rearranged equations will be identical: . 

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 and form a system, and is a multiple of the second equation, then the equations and form an equivalent system that can help us eliminate and make progress toward finding the value of .

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.

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).

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 to rely 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?"