Lesson 24

This warm-up reminds students about systems of equations and their solutions. Students recall that a solution to a linear equation in two variables is any pair of numbers that makes the equation true, and that a solution to a system of two equations in two variables is a pair of numbers that make both equations true.

The given system has a solution that is hard to find mentally, but can be calculated algebraically or by using graphing technology.

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

Students who graph the system of equations using technology may estimate from the graph and offer \((8,\text-2)\) as a solution. Ask them to check whether \(8+\text-2\) really does equal 5.678.

Ask students using different methods to briefly describe their solving process. Record and display their reasoning (including a graph) for all to see.

If not mentioned in students' explanations, highlight that the riddle can be solved by writing and solving a system of equations. Each equation represents a constraint. Ask students:

• "What constraints do the two equations represent?" (The first equation represents a constraint about the sum of the two numbers. The second represents a constraint about the difference of the two numbers.)
• "What does a solution to the system represent?" (The solution is a pair of numbers that simultaneously meet both constraints or make both equations true.)
• "How many pairs of numbers meet both constraints at the same time?" (Only one pair. The graphs of the equations intersect at one point.)

In this activity, students encounter a situation in which two constraints that can be expressed with inequalities are represented on two separate graphs, which makes it challenging to find pairs of values that meet both constraints simultaneously. This motivates a desire to represent both constraints on the same graph.

As students work, monitor for the different ways students try to find a pair of values that satisfy both inequalities. Some likely approaches:

• Substituting different \(x\)- and \(y\)-values into the inequalities until they find a pair that make both inequalities true.
• Visually estimating a point in the solution region of one graph that appears likely to also be in the solution region of the other graph (for example, noticing that \((6,4)\) is in the solution region of each inequality.
• Graphing both inequalities on the same coordinate plane (either using technology or by copying the line in one graph onto the other graph) and finding a point in the region where they overlap.

Identify students who use these or other approaches and ask them to share during class discussion.

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

Give students a moment to read the task statement and look at the two graphs. Ask students:

• “What are the two constraints in this situation?” (A length constraint and a cost constraint)
• “Which graph represents which constraint? How do you know?” (The first graph represents the length constraint. Possible explanations:
  • The graph intersects the vertical and horizontal axes at approximately 9.5, which means that if the quilter bought 0 yards of one color, he will need at least 9.5 yards of the other color.
  • The length constraint says "at least 9.5 yards," so the lengths must include values greater than 9.5, which is shown by the shaded region of the first graph.
  • The second graph represents the cost constraint. If the quilter bought 0 yards of the light color, he could buy up to \(\frac{110}{13}\), or about 8.5, yards of the dark color fabric. This corresponds to the vertical intercept of the second graph.) 
  • The cost constraint says "up to $110," so the lengths must be below certain limits. The solution region of the second graph shows values below a boundary.)

Arrange students in groups of 2 and provide access to graphing technology, in case requested.

Ask students to pause briefly after they have written both inequalities. Verify that the inequalities that students have written accurately represent the constraints before they proceed to the rest of the activity.

Focus the discussion on the last two questions. Select previously identified students to share how they identified a pair of values that meet both constraints. Sequence their presentation in the order listed in the Activity Narrative, which is from less systematic to more systematic.

Select students who graphed both inequalities on the same plane to display their graphs for all to see.

If no students did this on their own, display the embedded applet in the online materials for all to see. 

Consider selecting first one inequality, then the other, then both simultaneously. 

(The Desmos graphs show both variables being restricted to positive values, but students are not expected to do this when graphing. Consider discussing with students why it makes sense in this situation to disregard negative values.)

Discuss the meaning of a pair of values that satisfy both inequalities. Emphasize that, in this situation, it refers to the amount of fabric of each color that meets both the length and cost requirements.

Explain to students that the two inequalities representing the constraints in the same situation form a system of linear inequalities. In a system of linear equations, the solutions can be represented by one or more points where the graphs of the equations intersect. The solutions to a system of inequalities are all points in the region where the graphs of the two inequalities overlap, because those points represent all pairs of values that make both inequalities true.

In this activity, students write systems of inequalities that represent situations and find the solutions by graphing. Students have encountered the same situations and constraints in a previous lesson, so the main work here is on thinking about each pair of constraints as a system, finding the solution region of each inequality, and identifying a point in the region where the graphs overlap as a solution to the system.

Arrange students in groups of 2–4. Explain to students that they will now revisit some situations they have seen and graphed in an earlier lesson. Assign one situation to each group member (or allow them to choose one situation) and ask them to answer the questions. Provide access to graphing technology.

Give students 5-6 minutes of quiet work time, and then time to discuss their responses and graph with their group.

Students may need reminding what “a solution to the system” would be in these specific contexts. (First find a point where the total money in the two accounts totals less than or equal to $600. Now make sure the money in the saving account is also at least $50.)

Much of the discussion would have happened in small groups. During the whole-class discussion, emphasize the meaning of a point in the region where two graphs of linear inequalities overlap. Make sure students understand that all the points in that region represent values that simultaneously meet both constraints in the situation.

If time permits, ask students: "Why does it make sense to think of the two inequalities in each situation as a system and find the solutions to the system, instead of only to individual inequalities?" (If both constraints in the situation must be met, then we need to find values that satisfy both inequalities.)

This optional activity reinforces the idea that the solutions to a system of inequalities can be effectively represented by a region on the graphs of the inequalities in the system. The activity is designed to be completed without the use of graphing technology.

Ask students to put away any devices.

Some students may have trouble interpreting the graph of the fourth system, wondering if a point in either of the shaded regions on the graph could be where an item is hidden. Ask them to pick a point on the graph and consider whether it satisfies the first inequality, and then whether it satisfies the second inequality. Remind them that a solution to a system needs to satisfy both.

Invite students to share their graphs and strategies for finding the solution regions. In particular, discuss how they found out which system had no solutions.

Remind students that a system of linear equations has no solutions if the graphs of the equations are two parallel lines that never intersect. Explain that a system of linear inequalities has no solutions if their regions are bound by two parallel lines and the solution region of each one is on the "outside" of the parallel lines, as is the case with the last given system.