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Lesson 12Systems of Equations

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.

12.1 Milkshakes

Diego and Lin are drinking milkshakes. Lin starts with 12 ounces and drinks \frac14 an ounce per second. Diego starts with 20 ounces and drinks \frac23 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.

12.2 Passing on the Trail

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 toward the lake. Han’s distance, d , from the parking lot can be expressed as d = \text-2.4t+4.8 , where t 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.

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

  2. What do the coordinates mean in this situation?

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

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

12.3 Stacks of Cups

A stack of n small cups has a height, h , in centimeters of h=1.5n+6 . A stack of n large cups has a height, h , in centimeters of h=1.5n+9 .

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

Lesson 12 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 \frac14  foot each day. Plant B starts at 3 ft tall and grows at a constant rate of \frac12  foot each day. We can write equations y = \frac14 x + 6 for Plant A and y = \frac12 x +3 for Plant B, where x represents the number of days after being planted, and y represents height. We can write this system of equations.

\begin{cases} y = \frac14 x + 6 \\ y = \frac12 x +3 \end{cases}

Solving a system of equations means to find the values of x and y 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 x and y values that make both equations true. Here is a graph for the bamboo example:

Two functions comparing the time in days and height in feet of Plant A and Plat B. There is an intersecting point at (12, 9).

The solution to this system of equations is (12,9) , 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.

Glossary Terms

system of equations

A system of equations is a set of two or more equations. Each equation contains two or more variables. We want to find values for the variables that make all the equations true. 

These equations make up a system of equations:

x + y = \text-2
x - y = 12

The solution to this system is x = 5 and y = \text-7 because when these values are substituted for x and y , each equation is true:  5 + (\text-7) = \text-2 and 5 - (\text-7) = 12 .

Lesson 12 Practice Problems

  1. Here is the graph for one equation in a system of equations:

    A negative sloped function is graphed
    1. Write a second equation for the system so it has infinitely many solutions.
    2. Write a second equation whose graph goes through (0,1) so the system has no solutions.
    3. Write a second equation whose graph goes through (0,2) so the system has one solution at (4,1) .

  2. Create a second equation so the system has no solutions.

    \begin{cases} y=\frac34x -4 \\ \\ \end{cases}

  3. Andre is in charge of cooking broccoli and zucchini for a large group. He has to spend all $17 he has and can carry 10 pounds of veggies. Zucchini costs $1.50 per pound and broccoli costs $2 per pound. One line shows combinations of zucchini and broccoli that weigh 10 pounds and the other shows combinations of zucchini and broccoli that cost $17.

    A graph comparing the cost of zucchini in pounds and broccoli in pounds.
    1. Name one combination of veggies that weighs 10 pounds but does not cost $17.
    2. Name one combination of veggies that costs $17 but does not weigh 10 pounds.
    3. How many pounds each of zucchini and broccoli can Andre get so that he spends all $17 and gets 10 pounds of veggies?

  4. The temperature in degrees Fahrenheit, F , is related to the temperature in degrees Celsius, C , by the equation F = \frac{9}{5}C + 32

    1. In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?

    2. In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?

    3. There is one temperature where the degrees Fahrenheit and degrees Celsius are the same, so that C=F . Use the expression from the equation, where F is expressed in terms of C , to solve for this temperature.