Lesson 12Systems of Equations

Learning Goal

Let’s learn what a system of equations is.

Learning Targets

  • I can explain the solution to a system of equations in a real-world context.

  • I can explain what a system of equations is.

  • I can make graphs to find an ordered pair that two real-world situations have in common.

Lesson Terms

  • system of equations

Warm Up: Milkshakes

Problem 1

Diego and Lin are drinking milkshakes. Lin starts with 12 ounces and drinks an ounce per second. Diego starts with 20 ounces and drinks an ounce per second.

  1. How long will it take Lin and Diego to finish their milkshakes?

  2. Without graphing, explain what the graphs in this situation would look like. Think about slope, intercepts, axis labels, units, and intersection points to guide your thinking.

  3. Discuss your description with your partner. If you disagree, work to reach an agreement.

Activity 1: Passing on the Trail

Problem 1

There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake. Han and Jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times.

At the time that Han reaches the lake and starts to turn back, Jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour towards the lake. Han’s distance, , from the parking lot can be expressed as , where represents the time in hours since he left the lake.

  1. What is an equation for Jada’s distance from the parking lot as she heads toward the lake?

  2. Draw both graphs: one representing Han’s equation and one representing Jada’s equation. It is important to be very precise.

    open applet in presentation mode

  3. Find the point where the two graphs intersect each other. What are the coordinates of this point?

  4. What do the coordinates mean in this situation?

  5. What has to be true about the relationship between these coordinates and Jada’s equation?

  6. What has to be true about the relationship between these coordinates and Han’s equation?

Print Version

There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake. Han and Jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times.

At the time that Han reaches the lake and starts to turn back, Jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour towards the lake. Han’s distance, , from the parking lot can be expressed as , where represents the time in hours since he left the lake.

  1. What is an equation for Jada’s distance from the parking lot as she heads toward the lake?

  2. Draw both graphs: one representing Han’s equation and one representing Jada’s equation. It is important to be very precise! Be careful, work in pencil, and use a ruler.

    A blank graph of time (hours) (scale: 0-1.75) vs distance from parking lot (miles) (scale: 1-5).

  3. Find the point where the two graphs intersect each other. What are the coordinates of this point?

  4. What do the coordinates mean in this situation?

  5. What has to be true about the relationship between these coordinates and Jada’s equation?

  6. What has to be true about the relationship between these coordinates and Han’s equation?

Activity 2: Stacks of Cups

Problem 1

A stack of small cups has a height, , in centimeters of . A stack of large cups has a height, , in centimeters of .

  1. Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale.

    open applet in presentation mode

  2. For what number of cups will the two stacks have the same height?

Print Version

A stack of small cups has a height, , in centimeters of . A stack of large cups has a height, , in centimeters of .

  1. Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale.

    a blank grid

  2. For what number of cups will the two stacks have the same height?

Lesson Summary

A system of equations is a set of 2 (or more) equations where the variables represent the same unknown values. For example, suppose that two different kinds of bamboo are planted at the same time. Plant A starts at 6 ft tall and grows at a constant rate of foot each day. Plant B starts at 3 ft tall and grows at a constant rate of foot each day. We can write equations for Plant A and for Plant B, where represents the number of days after being planted, and represents height. We can write this system of equations.

Solving a system of equations means to find the values of and that make both equations true at the same time. One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point. The intersection point represents the pair of and values that make both equations true. Here is a graph for the bamboo example:

A graph of time (days) vs height (ft) with two lines, Plant A and Plant B. A starts at (0,6), B starts at (0,3) and they intersect at (12,9).

The solution to this system of equations is , which means that both bamboo plants will be 9 feet tall after 12 days.

We have seen systems of equations that have no solutions, one solution, and infinitely many solutions.

  • When the lines do not intersect, there is no solution. (Lines that do not intersect are parallel.)

  • When the lines intersect once, there is one solution.

  • When the lines are right on top of each other, there are infinitely many solutions.

In future lessons, we will see that some systems cannot be easily solved by graphing, but can be easily solved using algebra.