Meeting Constraints

2.1.3 Meeting Constraints

2.1.3 • Meeting Constraints

Activity

Use the following information to answer questions 1 – 3.

A dining hall had a total of 25 tables: some long rectangular tables and some round ones. Long tables can seat 8 people. Round tables can seat 6 people. On a busy evening, all 190 seats at the tables are occupied.

How many long tables $x$ are there?

20 long tables

How many round tables, $y$ , are there?

5 round tables.

Explain or show how you know how many different size tables there are.

Compare your answer:

Your answer may vary, but here is a sample.

$x + y = 25$
$8 x + 6 y = 190$
$x + y = 25$ can be rewritten as $y = 25 - x$ .

Substituting $25 - x$ for $y$ in $8 x + 6 y = 190$ gives $x = 20$ .

If $x$ is 20, then $y$ is 5.

Use the following information to answer questions 4 – 6.

A family bought a total of 16 adult and child tickets to a magic show. Adult tickets are $10.50 each, and child tickets are $7.50 each. The family paid a total of $141.

How many adult tickets, $a$ did they buy?

7 adult tickets

How many child tickets, $c$ did they buy?

9 child tickets.

Explain or show how you know the prices of the tickets.

Compare your answer:

Your answer may vary, but here is a sample.

$a + c = 16$
$10.50 a + 7.50 c = 141$

When the equations are graphed, the intersection of the two lines is at $(7, 9)$ .

Use the following information to answer questions 7 - 9

At a poster shop, Han paid $16.80 for 2 large posters and 3 small posters of his favorite band. Kiran paid $14.15 for 1 large poster and 4 small posters of his favorite TV shows. Posters of the same size have the same price.

Find the price of a large poster, $l$ .

$4.95 for a large poster.

Find the price of a small poster, $s$ .

$2.30 for a small poster.

Explain or show how you know the prices of the posters.

Compare your answer:

Your answer may vary, but here is a sample.

When the equations

$2 ℓ + 3 s = 16.80$ and $ℓ + 4 s = 14.15$ are graphed, the intersection of the two lines is at $(4.95, 2.30)$ .

Are you ready for more?

Extending Your Thinking

Make up an equation for $x$ whose two lines that intersect at $(4, 1)$ .

Compare your answer:

$x = 4$

Make up an equation for $y$ whose two lines that intersect at $(4, 1)$ .

Compare your answer:

$y = 1$

Make up equations for three lines whose intersection points form a triangle with vertices at $(- 4, 0)$ , $(2, 9)$ , and $(6, 5)$ .

Compare your answer:

Your answer may vary, but here is a sample.

$y = 32 x + 6$ , $y = - x + 11$ , $y = 12 x + 2$

Video: Meeting Constraints

Watch the following video to learn more about constraints.

Access multimedia content

Self Check

An automotive store is shipping car parts to a mechanic.

Each shock weighs 7 pounds.
Each brake pad weighs 4 pounds.

The store shipped 20 parts weighing a total of 104 pounds. How many shocks, $x$ , and brake pads, $y$ , did the mechanic buy?

Additional Resources

Meeting Constraints in a Real-World Situation

Let’s look at a situation similar to the activity and discuss how to find values that meet the constraints given.

THEATER SEATS

Auditorium seating chart with color-coded rows. Some rows are orange, indicating different seat types, while other rows are navy blue. "Screen" at the top.

A movie theater sells two types of tickets for a movie showing. Stadium recliner seats cost $18 each, and regular seats cost $15 each. The theater has a total of 40 seats. When every seat is sold, the theater earns $648 in ticket sales. How many stadium recliner seats, $x$ , and how many regular seats, $y$ , are there?

To solve this problem, we first need to recognize that there are two constraints in this situation. The constraints are the total number of seats in the theater and the total money earned from sales. Let’s write an equation to represent each constraint.

Constraint #1: the total number of seats

There are $x$ stadium recliner seats and $y$ regular seats representing a total of 40 seats.

This is represented by the equation $x + y = 40$ .

Constraint #2: the total sales from the seats

Stadium recliner seats cost $18 and regular seats cost $15, for a total of $648.

This is represented by the equation $18 x + 15 y = 648$ .

There are many ways to solve these equations: graphing and algebraic solving, guessing and checking, or using tables.

Try it

Meeting Constraints in a Real-World Situation

Let’s try creating a table. We know the total number of seats must equal 40. So we can try different values of seat types to determine if they match the total sales constraint, $648.

If the total sales value is too low, try more stadium recliner seats, since they are more expensive.

Number of Stadium Recliner Seats	Number of Regular Seats	Total Seats	Total Sales ($)
		40
		40
		40
		40

Here is how to fill in the table:

Number of Stadium Recliner Seats	Number of Regular Seats	Total Seats	Total Sales ($)
10	30	40	$630
12	28	40	$636
14	26	40	$642
16	24	40	$648

We can see that 16 stadium recliner seats and 24 regular seats represents a total of 40 seats and is $648 in total sales.

This means the theater has 16 stadium recliner seats and 24 regular seats.

How did you do? Know that you learn more from wrong answers than from right answers! Wrong answers help us find flaws in our thinking. Take a minute to think about why you got the question wrong. Then you won’t get this type of question wrong again!