Writing an Equation to Represent a Constraint

1.9.2 Writing an Equation to Represent a Constraint

1.9.2 • Writing an Equation to Represent a Constraint

Activity

An automobile manufacturer is preparing a shipment of cars and trucks on a cargo ship that can carry 21,600 tons.

The cars weigh 3.6 tons each, and the trucks weigh 7.5 tons each.

Write an equation that represents the weight constraint of a shipment. Let $c$ be the number of cars and $t$ be the number of trucks.

Compare your answer:

$3.6 c + 7.5 t = 21, 600$

For questions 2 – 5, use the following situation:

For one shipment, trucks are loaded first and cars are loaded afterward. (Even though trucks are bulkier than cars, a shipment can consist of all trucks as long as it is within the weight limit.)

Find the number of cars that can be shipped if the cargo already has 480 trucks.

5000

Find the number of cars that can be shipped if the cargo already has 1500 trucks.

2875

Find the number of cars that can be shipped if the cargo already has 2736 trucks.

300

Find the number of cars that can be shipped if the cargo already has $t$ trucks

Compare your answer:

$\frac{21, 600 - 7.5 t}{3.6}$

Use the following information for questions 6 – 8. For a different shipment, cars are loaded first, and then trucks are loaded afterward.

Write an equation you could enter into a calculator or a spreadsheet tool to find the number of trucks that can be shipped if the number of cars is known.

Compare your answer:

$t = \frac{21, 600 - 3.6 c}{7.5}$ , where $c$ is the number of cars.

Use your equation and a calculator or a computer to find the number of trucks that can be shipped if the cargo already has 1000 cars.

2400

Use your equation and a calculator or a computer to find the number of trucks that can be shipped if the cargo already has 4250 cars.

840

Are you ready for more?

Extending Your Thinking

For questions 1 and 2, use the following situation:

For yet another shipment, the manufacturer is also shipping motorcycles, which weigh 0.3 ton each.

Write an equation that you could enter into a calculator or a spreadsheet tool to find the number of motorcycles that can be shipped, $m$ , if the number of cars and trucks are known.

Compare your answer:

$m = \frac{21, 600 - 7.5 t - 3.6 c}{0.3}$

Use your equation to find the number of motorcycles that can be shipped if the cargo already contains 1,200 trucks and 3,000 cars.

Compare your answer:

6000

Self Check

The total revenue a high school makes from a play is T dollars. The revenue is made from selling tickets at $3.25 per child ticket, c, and $7.50 for each adult ticket, a. Which equation is best to find the number of child tickets sold?

$c= \frac{T}{3.25}$
$a= \frac{T-3.25c}{7.50}$
$T=3.25c+7.50a$
$C= \frac{T-7.50a}{3.25}$

Additional Resources

Write Constraint Resources

Review the following information from a previous lesson.

Writing Equations Using Symbols

When writing statements using algebraic symbols, the following steps may be helpful.

Step 1 - Read the problem.

Step 2 - Identify the variables and known values. If needed, sketch a picture of the scenario.

Step 3 - Write a sentence using the relationship among the values.

Step 4 - Translate the sentence into an equation.

Example

You are buying chicken and steak for a barbeque. Chicken costs $1.29/lb and steak costs $3.49/lb.

What variables might you need to use in this scenario?

Compare your answer: Your answer may vary, but here are some examples.

Let $c$ represent the number of pounds of chicken that you can buy.
Let $s$ represent the number of pounds of steak that you can buy.
Let $T$ represent the total cost of the barbeque.

What sentence might you use to describe the total cost of the barbeque?

Compare your answer: Your answer may vary, but here are some examples.

The total cost of the barbeque is the sum of the total cost of all the chicken purchased and the total cost of all the steak purchased.
The cost per pound of the chicken multiplied by the number of pounds purchased plus the cost per pound of the steak multiplied by the number of pounds purchased equals the total cost of the barbeque.

Write an equation to show the total cost, $T$ , of the meat.

Compare your answer:

$T = 1.29 c + 3.49 s$

You have $100 to spend on the barbeque. Use this information and rewrite the equation $T = 1.29 c + 3.49 s$ to find the number of pounds of chicken, $c$ , that you can buy.

Compare your answer:

$c = \frac{100 - 3.49 s}{1.29}$

$T = 1.29 c + 3.49 s$

$100 = 1.29 c + 3.49 s$

$100 - 3.49 s = 1.29 c + 3.49 s - 3.49 s$

$100 - 3.49 s = 1.29 c$

$\frac{100 - 3.49 s}{1.29} = \frac{1.29 c}{1.29}$

$\frac{100 - 3.49 s}{1.29} = c$

You have $100 to spend on the barbeque. Use this information and rewrite the equation $T = 1.29 c + 3.49 s$ to find the number of pounds of steak, $s$ , that you can buy.

Compare your answer:

$s = \frac{100 - 1.29 c}{3.49}$

$T = 1.29 c + 3.49 s$

$100 = 1.29 c + 3.49 s$

$100 - 1.29 c = 1.29 c - 1.29 c + 3.49 s$

$100 - 1.29 c = 3.49 s$

$\frac{100 - 1.29 c}{3.49} = \frac{3.49 s}{3.49}$

$s = \frac{100 - 1.29 c}{3.49}$

Try it

Write Constraint Equations

On a canoeing trip, you walk 4 mph then canoe at 7 mph and cover 30 miles. Write an equation to find how many miles you travel while canoeing.

Compare your answer:

Step 1 - Identify known variables. $c$ : miles canoeing $w$ : miles walked

Step 2 - Write an equation to show the distance covered. 4w+7c=30

Step 3 - Write an equation that solves for $c$ the number of miles traveled while canoeing. $c = \frac{30 - 4 w}{7}$