## Using Formulas to Solve Problems

### Learning Outcomes

• Set up a linear equation involving distance, rate, and time.
• Find the dimensions of a rectangle given the area.
• Find the dimensions of a box given information about its side lengths.

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Sometimes, these problems involve two equations representing two unknowns, which can be written using one equation in one variable by expressing one unknown in terms of the other. Examples of formulas include the perimeter of a rectangle, $P=2L+2W$; the area of a rectangular region, $A=LW$; and the volume of a rectangular solid, $V=LWH$.

### Recall the relationship between distance, rate and time

The distance $d$ covered when traveling at a constant rate $r$ for some time $t$ is given by the formula $d=rt$.

### Example: Solving an Application Using a Formula

It takes Andrew 30 minutes to drive to work in the morning. He drives home using the same route, but it takes 10 minutes longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

### How were the fractions handled in the example above?

Recall that when solving multi-step equations, it is helpful to multiply by the LCD to clear the denominators from the equation.

But it is also permissible to use operations on fractions to combine like terms. The example above demonstrates both.

### Try It

On Saturday morning, it took Jennifer 3.6 hours to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 hours to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

### Example: Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is $54$ ft. The length is $3$ ft. greater than the width. What are the dimensions of the patio?

### Evaluating a variable for an expression

In the example above, the length was written in terms of the width in order to substitute it for the variable $L$ to be able to write one equation in one variable.

This is a powerful technique in algebra and should be practiced until it becomes familiar, so it’s worth taking a closer look.

### Try It

Find the dimensions of a rectangle given that the perimeter is $110$ cm. and the length is 1 cm. more than twice the width.

### Example: Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in2. The length is $6$ in. more than the width. Find the area of the graph paper.

### Try It

A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft2 of new carpeting should be ordered?

### Example: Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $8$ inches, and the volume is 1,600 in.3.