The purpose of this Warm-up is to give students an intuitive and concrete way to think about combining two equations that are each true.

Students are presented with diagrams of three balanced hangers, which suggest that the weights on the two sides of each hanger are equal. Each side of the last hanger shows the combined objects from the corresponding side of the first two hangers. Students can reason that if 2 circles weigh the same as 1 square, and 1 circle and 1 triangle weigh the same as 1 pentagon, then the combined weight of 3 circles and 1 triangle should also be equal to the combined weight of 1 square and 1 pentagon.

Ask students to share the things that they noticed and wondered. Record and display, for all to see, their responses without editing or commentary. If possible, record the relevant reasoning on or near the diagrams. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

The idea to emphasize is that the weights on each side of the third hanger come from combining the weights on the corresponding sides of the first two hangers. If no one points this out, raise it as a point for discussion. Ask students:

• “What do you notice about the left side of the last hanger? What about the right side?"
• "If we saw only the first two hangers but knew that the third hanger has the combined weights from the corresponding side of the first two hangers, could we predict whether the weights on the third hanger would balance? "Why or why not?” (Yes. We can think of it as adding the weights from the second hanger to the first one, or vice versa. If the same weight is added to each side of a balanced hanger, the hanger would still be balanced.)

In the Warm-up of this lesson, students saw a visual representation of two equations being added to form a third equation. Because the first two equations are balanced, the third is also balanced. In this activity, students continue to develop the idea of adding two equations to form a third equation and use it to help them solve systems of linear equations.

Along the way, students examine the work of others and practice explaining their reasoning and critiquing that of others (MP3). They also see that sometimes adding equations is a productive way to solve systems, but other times it isn't.

Invite students to share their responses to the last set of questions, and discuss whether Diego's method works for solving the two systems. Ask students:

• “Why does adding the equations work for solving the system with and , but doesn't work for and ?” (In the latter system, the resulting equation still has two variables whose values we don't know.)
• “What if we subtract the equations? Would that help us solve the last system?” (Yes, subtracting the second equation from the first gives or , which we can then use to find the -value.)
• “How is adding the two equations here like and unlike combining the shapes in the two hanger diagrams earlier? How is it different?” (It is alike in that the result is another equation with the combined parts on the left side staying on the left and the combined parts on the right staying on the right. It is unlike the hanger diagrams in that these equations use numbers and variables, and in one of the systems, some of the parts on the left added up to 0 and are not shown in the third equation.)

Make sure students see that if we choose to add or subtract strategically, in each of the new equations, one variable is eliminated, making it possible to solve for the other variable. When the value of that variable is substituted to either of the original equations, we can solve for the variable that was eliminated. Tell students that this method of solving a system is called solving by elimination.

Point out that there is nothing wrong about adding the equations in the last system. It simply doesn't get us anywhere closer to the solution and is therefore unproductive.

Earlier, students saw that adding or subtracting the equations in a system creates a third equation that can help us solve the system. In this activity, students reconnect the idea of the solution to a system to the intersection of the graphs of the equations. They graph each original pair of equations and the equation that results from adding or subtracting them. They then observe that the graph of the third equation intersects the other two graphs at the exact same point—at the intersection of the first two.

At this point, students simply get a graphical confirmation that adding or subtracting equations can help them find the solution to a system. They are not yet expected to be able to articulate why this is the case. That understanding will be developed over a few upcoming lessons.

When solving System B, some students may not notice that the -variable in one equation has a positive coefficient and the other has a negative coefficient, and consequently decide to subtract the second equation from the first, rather than to add the two equations. They may struggle to figure out why the solution pair they find doesn't match what is on the graph. Suggest that they express the second equation in terms of addition, , and try eliminating one variable again.

Display, for all to see, the graphs that students generated, and ask students to share their observations. Highlight that the graph of the new equation intersects the graphs of the equations in the original system at the same point.

Time permitting, ask students to subtract the equations that they previously added (or to add the equations that they previously subtracted) and then to graph the resulting equation on the same coordinate plane. Ask them to comment on the graphs. Students are likely to see that the graphs of the new equations are no longer horizontal or vertical lines, but they still intersect at the same point as the original graphs.

Invite students to share their conjectures as to why the graph of the new equation intersects the other two graphs at the same point. Without confirming or correcting their conjectures, tell students that they will investigate this question in the coming activities.