standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Activity
The purpose of this activity is for students to practice graphing and making sense of the graphs of systems of equations. Students construct equations from descriptions of situations, graph those equations, then consider points on the graph in context. This will be useful in their Algebra class when students write their own system of equations to represent situations.
If students are proficient at graphing lines, consider providing access to graphing technology to focus student work on the creation of the equations and interpretation of the results.
Student Facing
A ride sharing company offers two options: riding in small cars that can carry up to 3 passengers each, or riding in large vans that can carry up to 6 passengers each. A group of 27 people is going to use the ride sharing service to take a trip. The trip in a small car costs
$
10 and the trip in a large van costs
$
15. The group ends up spending
$
80 total.
An equation that represents one of the constraints is
\(3x+6y=a\)
.
What is the value of
\(a\)
?
What do
\(x\)
and
\(y\)
represent?
An equation that represents the other constraint is
\(cx+15y=80\)
. What is the value of
\(c\)
?
[IMG]
Here is a graph that represents one of the constraints. Which one? Explain how you know.
Sketch another line on the graph that represents the other constraint.
For each coordinate pair, describe its meaning in the situation and decide whether it satisfies the constraint on total number of people, the constraint on cost, or neither.
\((2,4)\)
\((1,4)\)
\((3,2)\)
At what point do the two lines intersect? What does this point mean in this situation?
Student Response
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Activity Synthesis
The purpose of the discussion is to understand the connection between the situation and the equations as well as understanding the meaning of points on one of the lines, on both of the lines, or on neither of the line in context.
Display the graph showing both lines as well as the points from the activity.
[IMG]
Ask students, "How can the graph help you determine whether a point will fit one of the situations, both of the situations, or neither? How can the equations help you determine the same thing?" (On the graph, we can see if a point is on one of the lines, on both of the lines, or on neither of the lines. If it is on a line, it fits with that situation. From the equations, we can substitute in the
\(x\)
- and
\(y\)
-values to determine whether the values make the equation true or not. If the equation is true, the values fit with that situation.)


aligns: no