Rearranging Formulas

Example For a rectangle with length $l$ and width $w$ in centimeters, the area in square centimeters is $A = lw$ and the perimeter in centimeters is $P = 2l + 2w$, where $l > 0$ and $w > 0$. A temperature of $C$ degrees Celsius is $F$ degrees Fahrenheit, where

Example In $A = lw$ the subject is $A$. With $w$ as the subject the formula reads

Isolate the letter first and substitute numbers afterwards. With $A = 24$ and $l = 6$ this gives $w = 4$, and the same rearranged formula works for every other rectangle too.

Example Make $C$ the subject of $F = \frac{9}{5}C + 32$. The letter $C$ was multiplied by $\frac{9}{5}$ and then had $32$ added, so subtract $32$ first and then multiply by $\frac{5}{9}$,

As a check, $F = 212$ gives $C = \frac{5}{9} \cdot 180 = 100$, the boiling point of water.

A sum works the same way. To make $w$ the subject of $P = 2l + 2w$, subtract $2l$ and divide by $2$,

Because $w > 0$, this formula describes a real rectangle only when $P > 2l$.

These steps work because the chosen letter appears only to the first power. A formula such as $A = s^2$ for the area of a square needs a square root to make $s$ the subject, which is developed in Radicals and Rational Exponents once roots have been introduced.

Example In the motion formula $d = rt$, a car on one trip has a fixed speed $r$, so $r$ is a parameter while the time $t$ and the distance $d$ vary. At $r = 50$ miles per hour the formula is $d = 50t$, and after 3 hours the car has gone 150 miles.

Example Make $t$ the subject of the rate formula $d = rt$. Dividing by $r$ requires $r \ne 0$, so

For a moving car $r > 0$, and the restriction holds automatically. A trip of 150 miles at 60 miles per hour takes $t = \frac{150}{60} = 2.5$ hours. Similarly $w = \frac{A}{l}$ is safe for a rectangle, because a length is positive.

Example Solve $ax + 6 = 2x + b$ for $x$. Collect the $x$ terms and factor,

If $a \ne 2$, then

If $a = 2$, the equation reads $0 = b - 6$. It has no solution when $b \ne 6$, and every real $x$ is a solution when $b = 6$.

Factoring also helps in money formulas. With simple interest, a principal of $P$ dollars at an annual rate $r$ for $t$ years grows to $A = P + Prt$ dollars. Factoring out $P$ gives $A = P(1 + rt)$, and because $r \ge 0$ and $t \ge 0$, the factor $1 + rt$ is at least $1$, so

has no exceptional case. Solving for $r$ is different. It gives $r = \frac{A - P}{Pt}$, which needs $P \ne 0$ and $t \ne 0$. At $t = 0$ the formula says only $A = P$, and no rate can be read from it.