Previously, students saw that an equation in two variables can have many solutions because there are many pairs of values that satisfy the equation. This activity illustrates that idea graphically. Students see that the coordinates of all points on the graphs are pairs of values that make the equation true, which means that they are all solutions to the equation.

They also see that, because the given equation models the quantities and constraints in a situation, not all points on the graph are meaningful. For example, only positive or values on the graph (that is, only points in the first quadrant of the coordinate plane) have meanings in this context, because almonds and figs cannot have negative values for their weight.

During the activity, look for students who perform numerical computations straightaway and those who first write a variable equation and then use it to answer the first two questions.

Some students may say that the points not on the line are impossible given that Clare spent $75. Encourage these students to think about what those points would mean if we didn’t know how much money Clare spent.

Display the graph for all to see. Invite students to share their equation for the situation and their interpretations of the points on and off the graph. Make sure students understand that a point on the graph of an equation in two variables is a solution to the equation. Discuss questions such as:

• "What does the point mean in this situation?" (Clare purchased 10 pounds of almonds and 3 pounds of figs.)
• "Is that a possible combination of pounds of figs and almonds? Why or why not?" (No. It doesn't lie on the graph. Also, if Clare bought 10 pounds of almonds and 3 pounds of figs, it would cost her $87, not $75.)
• "From the graph, it looks like might be a solution, but it is hard to know for sure. Is there a way to verify?" (Substitute the values into the equation and see if they make the equation true.)
• "Suppose we extend the two ends of the graph beyond the first quadrant. Would a point on those parts of the line—say, —be a solution to the equation ? Why or why not?" (It would still be a solution to the equation, but it wouldn't make sense in this context. The weight of almonds or figs cannot be negative.)
• “What changes if the amount of figs is graphed on the -axis and the amount of almonds on the -axis?” (The order of the values in the coordinate pair would switch, so the slope and intercepts of the line would change.)

In the previous activity, students analyzed and interpreted points on a graph relative to an equation and a situation. In this activity, they write a linear equation to model a situation, use graphing technology to graph the equation, and then use the graph to solve problems. Each given situation involves an initial value and a constant rate of change.

Before students begin the activity, introduce them to the graphing technology available in the classroom. Offer a quick tutorial on how to graph equations, adjust the graphing window, and plot points. This tutorial could happen independently of the activity as long as it precedes the activity.

Select one group who analyzed the first situation and one group who analyzed the second situation, and ask them to share their responses. Display their graphs for all to see.

Focus the discussion on two things: the meanings of the points on the graph, and how the graph could be used to answer questions about the quantities in each situation. Discuss questions such as:

• “How did you find the answers to the first two questions?” (By calculation, for example, computing , or finding , and then dividing by 125.)
• “How did you find the answer to the last question?” (By calculation, for instance, finding , and then dividing by 125. Or, alternatively, by using the graph.)
• “Why do you think time is on the -axis?” (The way the problems are set up, time is the most useful input. For example, the student may want to know how much is in the account after a certain amount of weeks, and it may be useful to know how much water is left in the tank after a certain number of minutes.)

Highlight how the graph of the equations could be used to answer the questions. If not already mentioned by students, discuss how the graph of  can be used to find the answers to all the questions about the student's savings account, and the graph of  can help us with the questions about draining the water tank.

It is not essential that time be on the -axis. In particular, for the saving situation, it may be useful to have the amount in the account along the -axis to think about how long it would take to save a certain amount of money.

Keep the graphs of the two equations displayed for the Lesson Synthesis.