Activity
The Department of Streets of a city has a budget of $1,962,800 for resurfacing roads and hiring additional workers this year.
The cost of resurfacing a mile of 2-lane road is estimated at $84,000. The average starting salary of a worker in the department is $36,000 a year.
Write an equation that represents the relationship between the miles of 2-lane roads the department could resurface, , and the number of new workers it could hire, , if it spends the entire budget.
Compare your answer:
Use the equation you wrote in the first question to solve for .
Compare your answer:
Explain what the solution to the equation in question 2 represents in this situation.
Compare your answer:
It represents the number of new workers the department could hire if it resurfaces miles of roads and spends its entire budget.
Use the equation you wrote in the first question to solve for .
Compare your answer:
Explain what the solution to the equation in question 4 represents in this situation.
Compare your answer:
It represents the miles of roads the department could resurface if it spends the entire budget and hires new workers.
The city is planning to hire 6 new workers and use its entire budget. Which equation should be used to find how many miles of 2-lane roads it could resurface?
Explain your reasoning.
Compare your answer:
Substitute 6 for in the expression that is equal to to find the miles of roads that could be resurfaced.
To the nearest hundredth, how many miles of 2-lane roads could the city resurface if it hires 6 new workers?
20.8
Video: Writing an Inequality for the Constraint
Watch the following video for more help solving this problem.
Self Check
Additional Resources
Solve a Formula for a Specific Variable
Solving for a variable in a formula is actually finding equivalent equations.
To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.
Example 1
Solve the formula for when and .
Step 1 - Write the formula.
Step 2 - Substitute.
Step 3 - Divide, to isolate .
Step 4 - Simplify.
Example 2
Solve the formula for .
Step 1 - Write the formula.
Step 2 - Substitute.
(step not needed)
Step 3 - Divide, to isolate .
Step 4 - Simplify.
Example 3
Solve the formula for , when and .
Step 1 - Write the formula.
Step 2 - Substitute.
Step 3: Multiply to clear the fraction.
Step 4 - Simplify.
Step 5 - Solve for .
Example 4
Solve the formula for .
Step 1 - Write the formula.
Step 2 - Substitute.
(step not needed)
Step 3 - Multiply to clear the fraction.
Step 4 - Simplify.
Step 5 - Solve for .
Example 5
Solve the formula for when .
Step 1 - Write the formula.
Step 2 - Substitute.
Step 3 - Simplify.
Step 4 - Isolate the -term using subtraction.
Step 5 - Simplify.
Step 6 - Divide.
Step 7 - Simplify.
Example 6
Solve the formula for .
Step 1 - Write the formula.
Step 2 - Substitute.
(step not needed)
Step 3 - Simplify.
(step not needed)
Step 4 - Isolate the -term using subtraction.
Step 5 - Simplify.
Step 6 - Divide.
Step 7 - Simplify.
Try it
Try It: Solve a Formula for a Specific Variable
Solve for .
Compare your answer:
Here is how to solve this equation for by isolating:
Step 1 - Use inverse operations to bring the term to the right side of the equation by adding 4y.
Step 2 - Simplify.
Step 3 - Divide both sides by 2.
Step 4 - Simplify.