Lesson 12

Systems of Equations

Let’s learn what a system of equations is.

Problem 1

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

Graph of a line in the x y plane.
  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 \((4,1)\).

Problem 2

Create a second equation so the system has no solutions.

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

Problem 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 graph shows combinations of zucchini and broccoli that weigh 10 pounds and the other shows combinations of zucchini and broccoli that cost $17.

The graph of two intersecting lines in the x y plane.
  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?
(From Unit 4, Lesson 10.)

Problem 4

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

  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.

(From Unit 4, Lesson 9.)