Linear Equations, Inequalities, and Systems
In this unit, your student will analyze constraints on different quantities. For example, the amount you spend on a bicycle may be limited by how much you have saved. To qualify for a sports team, you may need to practice at least a certain number of hours, or lift at least a certain number of pounds.
Here are some ways to write constraints using mathematical notation:
\(w < 20\). An apartment building only allows dogs that weigh less than 20 pounds.
\(m + g + b = 4\). A casserole recipe calls for four cups of vegetables. You have mushrooms, green beans, and broccoli.
\(12.5c + 15a \geq 1,\!000\). In order for a concert to be performed, the artists need to be sure of $1,000 in ticket sales. Tickets for children under 18 are $12.50, and tickets for adults are $15.
\(5n+10d=150\). You need $1.50 in coins for a parking meter. You have a bunch of nickels and dimes in your pocket.
For this last situation, we can see that using more dimes to make $1.50 means that we can use fewer nickels, and vice-versa. A graph allows us to see this relationship even more clearly.
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 an overnight school trip. The cost of the trip is $360. She has a job at a convenience store, which pays $9 per hour, and sometimes babysits for a family in her neighborhood, which pays $12 per hour.
The equation \(9x+12y=360\) represents all the combinations of hours Priya could work at each job and earn a total of $360. Here is a graph showing those combinations:
- What are the coordinates of point \(A\)?
- What does it tell us about the number of hours Priya worked at each job?
- Answer the same questions about points \(B\) and \(C\).
- Point \(D\) is not on the line. How should we interpret point \(D\)?
- Point \(E\) is not on the line. How should we interpret point \(E\)?
- Priya works 20 hours at the convenience store and 15 hours babysitting.
- 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.
- Priya does not make enough money. She works 24 hours at the convenience store and 8 hours babysitting. She makes only $312, since \(24 \cdot 9 + 8 \cdot 12 = 312\).
- Priya makes more than enough money: $438. She works 30 hours at the convenience store and 14 hours babysitting. \(30 \cdot 9 + 14 \cdot 12 = 438\).