9.7 Solve a Formula for a Specific Variable

Formulas and Variables

A formula is a special type of equation that shows how different variables relate to each other. Solving a formula for a specific variable means rearranging it so that variable is alone on one side. This skill comes up constantly, whether you're finding the width of a rectangle when you know the area, or figuring out how long a road trip will take.

Distance, Rate, and Time Applications

The formula $d = rt$ connects three quantities: distance, rate (speed), and time. If you know any two, you can find the third.

• $d$ = total distance traveled (miles, kilometers)
• $r$ = rate or speed of travel (mph, kph)
• $t$ = time spent traveling (hours, minutes)

Finding distance (multiply rate by time):

• A car traveling at 60 mph for 2 hours covers $d = 60 \times 2 = 120$ miles

Finding rate (divide distance by time, since $r = \frac{d}{t}$):

• A train that travels 450 km in 3 hours has a rate of $r = \frac{450}{3} = 150$ kph

Finding time (divide distance by rate, since $t = \frac{d}{r}$):

• A train traveling 300 miles at 50 mph takes $t = \frac{300}{50} = 6$ hours

Notice the pattern: each version of the formula is just $d = rt$ rearranged to isolate a different variable. That rearranging process is exactly what this section is about.

Rearranging Formulas for Variables

The core idea is simple: whatever you do to one side of an equation, you must do to the other side. This keeps the equation balanced while you move things around to isolate the variable you want.

Moving terms with addition or subtraction:

If a term is added to your variable, subtract it from both sides (and vice versa).

• Rearrange $a + b = c$ to solve for $a$: subtract $b$ from both sides to get $a = c - b$
• Rearrange $x - 3 = 7$ to solve for $x$: add 3 to both sides to get $x = 10$

Removing factors with multiplication or division:

If your variable is multiplied by something, divide both sides by that factor. If it's divided by something, multiply both sides.

• Rearrange $xy = z$ to solve for $x$: divide both sides by $y$ to get $x = \frac{z}{y}$
• Rearrange $\frac{a}{4} = b$ to solve for $a$: multiply both sides by 4 to get $a = 4b$

Step-by-step example with a real formula:

Solve the perimeter formula $P = 2l + 2w$ for $w$.

1. Subtract $2l$ from both sides: $P - 2l = 2w$

2. Divide both sides by 2: $\frac{P - 2l}{2} = w$

3. So $w = \frac{P - 2l}{2}$

This same two-step pattern (subtract, then divide) works any time a variable is multiplied by a coefficient and grouped with other terms.

Contextual Meaning of Formula Variables

Variables aren't just letters. Each one represents a real quantity, and understanding what it stands for helps you set up and solve problems correctly.

Geometry formulas use variables for dimensions and measurements:

• In $A = lw$, $A$ is area, $l$ is length, and $w$ is width
• In $V = \pi r^2 h$, $V$ is volume, $r$ is the radius of the circular base, and $h$ is height

Science formulas use variables for measurable quantities:

• $d$ for distance, $v$ for velocity, $a$ for acceleration, $t$ for time

Real-world formulas work the same way. For example, a rental car cost formula like $C = rt + f$ uses $C$ for total cost, $r$ for hourly rate, $t$ for rental time, and $f$ for a fixed fee. If you wanted to figure out how many hours you can rent the car for a given budget, you'd solve for $t$:

1. Subtract $f$ from both sides: $C - f = rt$

2. Divide both sides by $r$: $t = \frac{C - f}{r}$

When you encounter an unfamiliar formula, always start by identifying what each variable represents in the context of the problem. That way, you'll know which variable to solve for and whether your answer makes sense.