This Math Talk focuses on values meeting a constraint. It encourages students to think about using values in an equation and to rely on what they know about cost in context to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students write and solve systems of linear equations in two variables.

Many students are likely to approach the task by multiplying pounds of raisins by 4 and pounds of walnuts by 8, and then see if the sum of the two products is 15. Some students may arrive at their conclusions by reasoning and estimating. For example, seeing that a pound of walnuts costs \$8, students may simply reason that the first combination is not possible. Or they may reason that the last combination is impossible because at \$4 a pound, 3.5 pounds of raisins would cost more than \$12, so it is not possible to also get 1 pound of walnuts and pay only \$15 total. 

In describing their strategies, students need to be precise in their word choice and use of language (MP6).

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

Conclude by reminding students that the given situation involves a cost constraint. Emphasize that only the third option, 2.25 pounds of raisins and 0.75 pounds of walnuts, satisfy the constraint. One way to check if certain values meet the constraint is by writing an equation and checking if it is true. For example, the equation is true. If we replace the weights of raisins and walnuts with other pairs of values, the equation would be false.

In this activity, students write and graph equations to represent two constraints in the same situation and use tables and graphs to see possible values that satisfy the constraints. The work prompts them to think about a pair of values that simultaneously meets multiple constraints in the situation, which in turn helps them make sense of the phrase "a solution to both equations."

Monitor for students who use these strategies:

• Guessing and checking
• Using the graph (by identifying the point with a given - or -value and estimating the unknown value, or using technology to get an estimate)
• Substituting the given value of one variable into the equation and solving for the other variable

Plan to have students present in order from less to more precise.

Some students might think that the values in the second table need to reflect a total cost of \$15. Clarify that the table represents only the constraint that Diego bought a total of 2 pounds of raisins and walnuts.

The idea of finding an pair that satisfies multiple constraints should be familiar from middle school. If students struggle to answer the last question, ask them to study the values in the table. Ask questions such as: "If Diego bought 0.25 pound of raisins, would he meet both the cost and weight requirements?" and "Which combinations of raisins and walnuts would allow him to meet both requirements? How many combinations are there?"

Invite previously selected students to share their response and reasoning for the last question. Sequence the discussion of the strategies by the order listed in the Activity Narrative. Display the graphs representing the system (either created by a student or as shown in the Student Response).

Connect the different responses to the learning goals by asking questions such as:

• “Can you use your strategy to find other combinations of raisins and walnuts, besides 0.25 pound and 1.75 pounds, that meet both cost and weight constraints?” (No)
• “How many possible combinations of raisins and walnuts meet both constraints? How do we know?” (One combination, because the graphs intersect only at one point.)

Explain to students that the two equations written to represent the constraints form a system of equations. We use a curly bracket to indicate a system, like this: 

Highlight that the solution to the system is a pair of values (in this case, pounds of raisins and walnuts) that meet both constraints. This means the pair of values is a solution to both equations. Graphing is an effective way to see the solution to both equations, if one exists.

This activity allows students to apply what they learned about meeting multiple constraints simultaneously and to practice solving simple systems of equations in context. Students are given several problems that can be efficiently solved by writing and solving systems.

As students work, notice the strategies they use. Many students may choose to graph the systems because that strategy was used in the first activity, but some students may choose to solve the systems by guessing and checking, creating tables, or by reasoning algebraically. (In middle school, students learned the substitution method for solving systems of equations. Expect students to have varying degrees of facility with it at this point.)

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

If students struggle to sort out the information in a problem, suggest that they start by drawing a picture or diagram to help understand the situation. Another idea is to start by creating one or more tables to list possible combinations of values and observe the relationship between the values. 

For students who need more scaffolding, ask questions such as “What equation can you write to say that the number of tables, long tables and round ones, is 25?” and “What do your variables represent?” Then, ask students to use the same variables to write an equation that says that there are 190 seats.

Select students who use different strategies to share their responses and reasoning. Start with strategies that are less systematic (such as guessing and checking) and move toward those that are more systematic (graphing or using algebra). If possible, display their work for all to see. After each student presents, ask if others solved it the same way.

If no students share an algebraic strategy, ask if anyone solved the problems without graphing. If so, invite them to share their rationale. Otherwise, it is not crucial to probe further at this moment.

If time permits, ask students to reflect on the strategies that were used. Some of them—graphing and algebraic solving—required writing equations. Others—guessing and checking, or using tables—did not. Discuss questions such as:

• “Why might it be helpful to write a pair of equations to solve these problems?”
• “Are there situations in which it might not be helpful to write equations?”