Lesson 13
Systems of Equations
Problem 1
Here is the graph for one equation in a system of equations:
![Graph of a line in the x y plane.](https://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=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202829Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f4dda21991507146a0be17e857dcc0996973ed5e3db974ec8fab13016b1083b2)
- 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 \((4,1)\).
Solution
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Problem 2
Create a second equation so the system has no solutions.
\(\begin{cases} y=\frac34x -4 \\ \\ \end{cases}\)
Solution
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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://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=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202829Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=24162aa95e3ae6c59d91082e2c44991a4906890b9559e200c2979541358273f6)
- 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?
Solution
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(From Unit 5, Lesson 12.)Problem 4
The temperature in degrees Fahrenheit, \(F\), is related to the temperature in degrees Celsius, \(C\), by the equation \(\displaystyle F = \frac{9}{5}C + 32\)
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In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?
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In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?
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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.
Solution
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(From Unit 4, Lesson 17.)