Lesson 13
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.](https://staging-cms-im.s3.amazonaws.com/Su2Bb6aVEFnJhRzaGoL4xiBd?response-content-disposition=inline%3B%20filename%3D%228-8.4.C12.PP.line4.png%22%3B%20filename%2A%3DUTF-8%27%278-8.4.C12.PP.line4.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082355Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d5262f4ea834062a58a7db8ecafe6765e5b221d67935fe8599baeede743d7a0c)
- Write a second equation for the system so it has infinitely many solutions.
- Write a second equation whose graph goes through \((0,1)\) so the system has no solutions.
- 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.](https://staging-cms-im.s3.amazonaws.com/pvriHVLKQnfjRdXV5V8AkdJ5?response-content-disposition=inline%3B%20filename%3D%228-8.4.C10.PP.broccoli.png%22%3B%20filename%2A%3DUTF-8%27%278-8.4.C10.PP.broccoli.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082355Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a09d730e12dcb940a17a83f5ce8448b0e248e016884246e7b055017c2f8464b7)
- Name one combination of veggies that weighs 10 pounds but does not cost $17.
- Name one combination of veggies that costs $17 but does not weigh 10 pounds.
- How many pounds each of zucchini and broccoli can Andre get so that he spends all $17 and gets 10 pounds of veggies?
Problem 4
The temperature in degrees Fahrenheit, \(F\), is related to the temperature in degrees Celsius, \(C\), by the equation \(\displaystyle F = \frac{9}{5}C + 32\)
-
In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?
-
In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?
-
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.