In this activity, students start with an equation relating distance and time for Han’s hike and enough information to write a second equation relating distance and time for Jada’s hike. After writing Jada’s equation and graphing both lines, students then use the lines to identify the point of intersection and make sense of the point’s meaning in the context.

This activity is a culmination of students’ work writing, solving, and graphing equations along with the thinking that they have done on what it means for an equation to be true. From this foundation, students are ready to understand solving systems of equations from an algebraic standpoint in the following lessons. Fluently solving systems algebraically is not expected at this time.

In the digital version of the activity, students use an applet to graph the distance of Han and Jada from the parking lot over time. The applet allows students to precisely graph the lines and find the point of intersection. Use the digital version if students have difficulty graphing precisely enough to find an accurate point of intersection.

The purpose of this discussion is to strengthen the connection between graphs and equations and to formally introduce the vocabulary for systems of equations.

Begin the discussion by asking groups to share the intersection point.

Display a graph of the two equations for all to see alongside the system of equations:

Explain to students that this is called a system of equations, and that “solving a system of equations” means to find the values of the variables that make both equations true at the same time. Point out that in this problem, the solution to the system of equations is the point where Han and Jada are at the same distance from the parking lot at the same time (point to ).

Tell students that it is also possible to solve a system of equations without graphing. Because they will be the same distance from the parking lot at the same time, the values for will be equal. Because they both have an expression that is equal to , we can set those expressions equal to one another to get the equation . Solving this equation for will give the value for when they are the same distance from the parking lot at the same time.

Ask students to solve this equation and confirm that , which is the same value they found earlier by carefully graphing the lines of each equation. Emphasize that the intersection point gave a value of both and . To get the same information from the equations, it is important to substitute back into one of the equations to find the value for . Because Han and Jada are at the same distance from the parking lot when , it doesn't matter which equation is used to find the value of .

Students explore a system of equations with no solutions, in the familiar context of cup stacking. The context reinforces a discussion about what it means for a system of equations to have no solutions, both in terms of a graph and in terms of the equations (MP2). Over the next few lessons, the concept of one solution, no solutions, and infinitely many solutions will be abstracted to problems without context. In those situations, it may be useful to refer back to the context in this activity and others as a way to guide students toward abstraction.

In the digital version of the activity, students use an applet to graph the heights of two stacks of cups. The applet allows students to precisely graph the lines to see that they never intersect. Use the digital version if students have difficulty graphing precisely enough to show that the lines do not intersect or if appropriate tools for graphing are not readily available.

The key point for discussion is to connect what students observe about the graph to the idea (from earlier lessons) that equations can have no solution. Graphically, students see that the lines are parallel and are always separated by a distance that represents 3 cm. This means that a  stack of n large cups will always be 3 cm taller than a stack of n small cups. Connecting this to the equations, this means that there is no value of n that is a solution to both and at the same time. By the end of the discussion, students should understand that these statements about this situation are equivalent:

• The lines don't intersect.
• The lines are parallel.
• There is no value of n for which the stacks have the same height.
• There is no value of that makes true.

Invite students to explain how they used the graph or equations to answer the second question. Ask other students if they answered the question with a different line of reasoning. If not brought up by students, demonstrate that setting the expression for the height of the large cup, , equal to the expression for the height of the small cup, , and subtracting from both sides gives , which is false no matter what value of is used.