This activity serves two goals. The first goal is to further build students' intuition about the new variable equation that comes from adding two variable equations in a system, and about the solution to that new equation. This is done by grounding the addition and the sum in a familiar context.

The second goal is to support students in reasoning about why the new equation shares a solution with the original system. The context gives students a concrete mental reference, which can be helpful for interpreting the intersection of the graphs of all three equations and for thinking about the solution that all three equations share.

In this activity, students also encounter a system that they can solve by graphing and by substitution, but that cannot be easily solved simply by adding or subtracting the equations. This observation may pique students' curiosity and make them wonder if it is possible to solve all systems by elimination.

As students work, identify students who solve the system by graphing and those who solve by substitution. Ask them to share their work later.

Select previously identified students to share their solution and strategy for solving the system. Display their work (especially student-generated graphs) for all to see, or consider displaying this graph:

Emphasize that 21.50 and 6.50 are the unit prices of the two supplies that make both equations true.

Next, discuss students' responses to the last set of questions. Ask questions such as:

• "The first equation was . We added to the left side and 178.50 to the right side to get . Are the two sides of the resulting equation still equal? Why or why not?" (Yes. We’re adding equal amounts to each side, so the two sides are still equal. The cost of buying 8 calculators and 1 measuring tape is $178.50.)
• "Predict what the graphs of all three equations would look like." (They would intersect at a common point.) Consider graphing all three equations for all to see.
• "In this situation, why does it make sense for the graphs to intersect at the same point? What does that point mean?" (The prices of a calculator and of a measuring tape in the first two orders are also the prices in the combined order. The unit prices of the items didn’t change, so it makes sense that they work for all three equations.)
• "In general, why does it make sense for a third equation that is the sum of the two equations in a system to share a common solution with the other two?" (Adding the two equations means adding equal values to each side of one equation, which keeps the two sides of the sum equal. If there is a pair of and values that makes the first two equations true, that pair also makes the third equation true. )
• "Were you able to solve the system by adding or subtracting the equations, without graphing? Why or why not?" (No, because no variables get eliminated when we add or subtract.)

In this activity, students practice using algebra to solve systems of linear equations in two variables and checking their solutions. Students do not have to use elimination, but the equations in the first three systems conveniently have opposites for the coefficients of one variable, so one variable can be easily eliminated.

In the last system, none of the coefficients of the variables are opposites. Some students may choose to solve the system by substitution, but the process would be pretty cumbersome. The complication that students encounter here motivates the need for another move, which students will explore in the next lesson.

Students who opt to use technology to check their solutions practice choosing tools strategically (MP5).

Invite students to share their solutions and strategies for the first three systems and how they check their solutions. Then, focus the discussion on the last system. Solicit the strategies that students used for approaching that system. If someone solved it by substitution, display the work for all to see. If no one did, ask if it is possible to do. (If time permits, consider asking students to attempt to do so, or demonstrating that strategy to illustrate that it is not exactly efficient.)

Discuss questions such as:

• “What happens if we add or subtract the equations?” (No variables are eliminated, so it doesn’t help with solving.)
• “Why were you able to solve the first three systems by elimination but not the last one?” “What features were in the equations in those systems but not in the last one?” (In the first three systems, at least one variable in each pair of equations had the same or opposite coefficients, so when the terms were added or subtracted, the result was 0.)

We need new moves! Tell students that in an upcoming lesson they will explore another way to solve a system by elimination.