2.6 Solve a Formula for a Specific Variable

Solving Formulas for Specific Variables

A formula is an equation that shows the relationship between two or more variables. You already know formulas like $d = rt$ for distance or $A = lw$ for the area of a rectangle. In this section, you'll learn how to rearrange these formulas to solve for any variable you need. The process is the same as solving a regular equation: use inverse operations to get the variable you want by itself.

Distance, Rate, and Time

The formula $d = rt$ connects distance ($d$), rate or speed ($r$), and time ($t$). As written, it solves for distance. But you can rearrange it to solve for either of the other two variables.

Solving for rate: Divide both sides by $t$:

$r = \frac{d}{t}$

Solving for time: Divide both sides by $r$:

$t = \frac{d}{r}$

For example, if a car travels 120 miles in 2 hours:

• Rate: $r = \frac{120}{2} = 60$ mph
• If you know the rate is 60 mph and the distance is 120 miles: $t = \frac{120}{60} = 2$ hours

This same formula applies to any motion scenario, whether it's a train, a runner, or an airplane.

How to Isolate a Variable

Solving a formula for a specific variable means rearranging it so that variable is alone on one side. You treat every other letter as if it were a number, then use inverse operations just like you would in a regular equation.

Steps:

1. Identify which variable you're solving for.

2. Use inverse operations to undo whatever is being done to that variable.

   • Addition and subtraction undo each other.
   • Multiplication and division undo each other.

3. Work to get the target variable by itself on one side of the equation.

Example: Solve $A = \frac{1}{2}bh$ for $h$.

1. Multiply both sides by 2 to clear the fraction: $2A = bh$
2. Divide both sides by $b$ to isolate $h$: $h = \frac{2A}{b}$

Example: Solve $P = 2l + 2w$ for $l$.

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

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

Notice that in each case, you're doing the same thing you'd do if the other variables were just numbers. That's the key idea.

Common Formulas in Geometry and Science

You'll often need to rearrange formulas from geometry and science. Here are some common ones:

Geometry:

• Rectangle area: $A = lw$
• Triangle area: $A = \frac{1}{2}bh$
• Circle circumference: $C = 2\pi r$
• Perimeter of a rectangle: $P = 2l + 2w$

Science:

• Density: $\rho = \frac{m}{V}$ ($\rho$ = density, $m$ = mass, $V$ = volume)
• Force: $F = ma$ ($F$ = force, $m$ = mass, $a$ = acceleration)

Example: Solve the density formula for mass.

1. Start with $\rho = \frac{m}{V}$
2. Multiply both sides by $V$: $m = \rho V$

Example: Solve $F = ma$ for $a$.

1. Divide both sides by $m$: $a = \frac{F}{m}$

Formula Transformation Techniques

A few tips that help when formulas get more complex:

• Clear fractions first. If the formula has a fraction, multiply both sides by the denominator. This makes the rest of the work much simpler.
• Move terms without your target variable to the other side. Use addition or subtraction to get all terms that don't contain your variable away from it.
• Divide (or multiply) last. Once your variable's term is isolated, divide or multiply to get the variable completely alone.
• Substitute known values after rearranging. It's usually easier to solve for the variable first, then plug in numbers, rather than substituting early and working with a messy equation.

The more you practice rearranging formulas, the more natural it becomes. Every formula follows the same rules of algebra you've already been using.