Literal Equations and Formulas
Rearrange formulas to isolate a specified variable using the same inverse-operation techniques used for numerical equations. Apply to physics (d = rt), geometry (A = lw), and temperature conversion formulas.
A lesson of Algebra 1, in Solving Linear Equations, about 25 minutes.
By the end of this lesson, you will be able to rearrange any formula to make a chosen variable the subject, using the same inverse operations you use on numerical equations, including the case where the target variable appears in more than one term.
A Formula Is an Equation Wearing Letters
A literal equation is an equation with two or more variables in it. Every formula you have ever used is one.
$$d = rt \qquad A = lw \qquad F = \frac{9}{5}C + 32$$
$$d = rt$$
is convenient when you know the rate and the time and want the distance. But suppose you know the distance and the time and want the rate. The same fact, rearranged, is what you need:
$$r = \frac{d}{t}$$
Rearranging is not a new skill. It is solving an equation, with one change of mindset: treat every letter except your target as if it were a number.
You are not going to find out what $t$ is, and you do not need to. $t$ behaves exactly like $7$ would, so divide by it exactly as you would divide by $7$.
Where You Will Actually Use This
| Subject | Formula | Often rearranged for |
|---|---|---|
| Physics | $v = v_0 + at$ | $a$, the acceleration |
| Chemistry | $PV = nRT$ | $T$, the temperature |
| Geometry | $A = \frac{1}{2}bh$ | $h$, the height |
| Finance | $A = P(1 + rt)$ | $r$, the rate |
The SAT asks several of these every year, and every science course assumes you can do it without being taught again.
The Approach
- Decide which letter you are isolating. That one is the unknown
- Every other letter is a constant. Treat it like a number
- Undo the operations attached to your target, in reverse order
- If the target appears in more than one term, factor it out first
Example 1: One Multiplication to Undo
Here is the whole method on one formula, worked all the way through. Make $r$ the subject of $d = rt$. $r$ is being multiplied by $t$, so dividing by $t$ is what frees it.
- Divide both sides by t, writing both sides in full d/t = rt/t
- Simplify, and write the target on the left r = d/t
Worked Example 1: Your Turn
Make $w$ the subject of $A = lw$, the area of a rectangle.
Example 2: Two Things to Undo
Make $h$ the subject of $A = \dfrac{1}{2}bh$, the area of a triangle. $h$ has been multiplied by $b$ and by $\dfrac{1}{2}$, so both have to come off. Clearing the fraction first keeps it tidy.
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