In this unit, students analyze equations using variables to represent unknown values. For example, a recipe may call for 4 cups of vegetables. If you are going to use mushrooms (\(m\)), green beans (\(g\)), and broccoli (\(b\)), you might write \(m + g + b = 4\) to represent the number of cups of each vegetable you plan to use.

\(5n+10d=150\) may represent the number of dimes and nickels you could use to pay \\$1.50 at a parking meter. For this situation, we can see that using more dimes to make \\$1.50 means that we can use fewer nickels, and vice-versa.

Each point on the graph represents a combination of nickels and dimes that totals $1.50. For example, if you use 8 nickels, you will need 11 dimes.

Here is a task for you to try with your student:

Priya is saving money to go on a trip. The cost of the trip is \\$360. She has a job at a convenience store, where she earns \)9 per hour, and she sometimes babysits for a family in her neighborhood, for which she earns $12 per hour.

The equation \(9x+12y=360\) represents all the combinations of hours Priya could work at each job to earn a total of \\$360. Here is a graph showing those combinations:

Solution:

1. \((20,15)\)
2. Priya works 20 hours at the convenience store and 15 hours babysitting.
3. Point \(B\): \((32,6)\). Priya works 32 hours at the convenience store and 6 hours babysitting. Point \(C\): \((40,0)\). Priya works 40 hours at the convenience store and does not babysit at all.
4. Priya does not make enough money. She works 24 hours at the convenience store and 8 hours babysitting. She has made only \\$312, because \(24 \cdot 9 + 8 \cdot 12 = 312\).
5. Priya makes more than enough money: \\$438. She works 30 hours at the convenience store and 14 hours babysitting. Her total earnings are \(30 \cdot 9 + 14 \cdot 12 = 438\).