This Warm-up reminds students about the fact that some systems of linear equations have many solutions, while also prompting them to use what they learned in this unit to better understand what it means for a system to have infinitely many solutions.

The first equation shows two variables adding up to 5, so students choose a pair of values whose sum is 5. They notice that all pairs chosen are solutions to the system. Next, they try to find a strategy that can show that there are countless other pairs that also satisfy the constraints in the system. Monitor for these likely strategies:

• Solving by graphing: The graphs of the two equations are the same line, so all the points on the line are solutions to the system.
• Solving by substitution: The first equation can be rearranged to . Substituting for in the second equation gives , or , or . This equation is true no matter what is.
• Solving by elimination: If we multiply the first equation by 4, rearrange the second equation to , and then subtract the second equation from the first, the result is . Subtracting from each side gives , which is true regardless of what or is.
• Reasoning about equivalent equations: If we rearrange the second equation so that the variables are on the same side , we can see that this equation is a multiple of the first and that the two equations are equivalent. This means they have the exact same solution set, which contains infinite possible pairs of and .

Identify students who use different strategies, and ask them to share their thinking with the class later.

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

Ask the class how many solutions they think the system has. Select previously identified students to explain how they know that the system has infinitely many solutions, or that any pair of values that add up to 5 and make the first equation true also make the second equation true.

Students are likely to think of using graphs. It is not essential to elicit all strategies shown in the Activity Narrative, but if no one thinks of an algebraic explanation, be sure to bring one up. The last strategy mentioned in the Activity Narrative (reasoning about equivalent equations) is likely to be intuitive to students.

Make sure students see that the equations are equivalent, so all pairs of and values that make one equation true also make the other equation true, giving an infinite number of solutions.

In the Warm-up, students saw that some systems have infinitely many solutions. In this activity, they encounter a situation that can be represented with a system of equations but the system has no solutions. Students write equations to represent the two constraints in the situation and then solve the system algebraically and graphically.

As students work, notice the different ways in which they reach the conclusion that the systems have no solutions. Identify students with varying strategies, and ask them to share later.

In earlier activities, students gained some insights into the structure of equations in systems that have infinitely many solutions and those that have no solutions. In this activity, they apply those insights to sort systems of equations based on the number of solutions (one solution, many solutions, or no solutions).

Students could solve each system algebraically or graphically and sort afterward, but given the number of systems to be solved, they will likely find this process to be time consuming. A more productive way would be to look for and make use of the structure in the equations in the systems (MP7), for example, by looking out for equivalent equations, equations with the same slope but different vertical intercepts, variable expressions with the same or opposite coefficients, and so on.

To effectively make use of the structure of the systems, students need to attend closely to all parts of each equation—the signs, variables, coefficients, and constants—and to rearrange equations with care (MP6).

As students discuss their thinking in groups, make note of the different ways they use structure to complete the task. Encourage students who are solving individual systems to analyze the features of the equations and see if they could reason about the solutions or gain information about the graphs that way.

In this activity, students are analyzing the structure of equations in the systems, so technology is not an appropriate tool.