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.

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.