Unit 2 Family Materials

Linear Equations, Inequalities, and Systems

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:

1. What are the coordinates of point $$A$$?
2. What does it tell us about the number of hours Priya worked at each job?
3. Answer the same questions about points $$B$$ and $$C$$.
4. Point $$D$$ is not on the line. How should we interpret point $$D$$?
5. Point $$E$$ is not on the line. How should we interpret point $$E$$?

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 makes only $312, since $$24 \cdot 9 + 8 \cdot 12 = 312$$. 5. Priya makes more than enough money:$438. She works 30 hours at the convenience store and 14 hours babysitting. $$30 \cdot 9 + 14 \cdot 12 = 438$$.