Equations with More Than One Variable

Main idea

Solving a formula for one of its letters uses exactly the same moves as solving any equation. Just pretend every other letter is a number you happen not to know yet.

A = P(1 + r{\htmlData{sym=0}{t}}) \implies {\htmlData{sym=0}{t}} = \frac{A - P}{Pr}

What the symbols mean

$t$
Target variable
Isolate it; everything else is treated as a constant.

Explanation

Solving for a variable is simply rearranging a formula so a different letter is on its own. The interest formula $A = P(1 + rt)$ tells you how much money $A$ you'll have; rearranged to $t = \frac{A - P}{Pr}$ , it tells you how long you'll wait. Same pieces, $A$ (the final amount), $P$ (the starting amount), $r$ (the rate) and $t$ (the time), just asking a different question. All those letters can look intimidating, but you already know every move involved.

Why it's nothing new. Solve $3t + 2 = 11$ and you subtract 2, then divide by 3. Solve $Pr\,t + P = A$ for $t$ and you subtract $P$ , then divide by $Pr$ . The steps are identical; the numbers just have names.

Steps. Get every term with the target variable on one side, factor it out if it appears more than once, then divide by whatever is multiplying it.

What it's good for. A rearranged formula answers "what input gives this output?" Solve $F = \frac95C + 32$ for $C$ and you've built a Celsius converter, good for every temperature at once.

See it

This is the conversion F = 1.8C + 32. Solving it for C gives the reverse, C = (F − 32)/1.8, and that's the same line, just read from the other axis. Notice it crosses the dashed line F = C at only one spot, so the two scales agree at exactly one temperature: −40°.

Worked examples

Example 1

Solve $A = P(1 + rt)$ for $r$ .

1
Distribute to get $r$ out of the parentheses: $A = P + Prt$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Solve $F = \frac95C + 32$ for $C$ and evaluate at $F = 212$ .

Problem 2+5 XP

Solve $A = \frac12bh$ for $h$ , with $A = 30$ and $b = 12$ .

Problem 3+5 XP

Solve $y = \frac{x + 1}{x - 2}$ for $x$ in terms of $y$ , then evaluate at $y = 4$ .

Problem 4+5 XP

Solve $V = \pi r^2h$ for $h$ with $V = 36\pi$ and $r = 3$ .

Problem 5+5 XP

Solve $\frac1R = \frac1{R_1} + \frac1{R_2}$ for $R$ with $R_1 = 6$ , $R_2 = 3$ .

Problem 6+5 XP

Kinetic energy is $E = \frac12mv^2$ . Solve for $v$ , then find $v$ when $E = 450$ J and $m = 4$ kg.

Problem 7+5 XP

Money grows as $A = P(1 + r)^t$ . Solve for $r$ , then find $r$ when $A = 1210$ , $P = 1000$ and $t = 2$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

The thin lens equation is $\frac1f = \frac{1}{d_o} + \frac{1}{d_i}$ . Solve it for $d_i$ , then find the image distance for focal length $f = 10$ cm and object distance $d_o = 15$ cm.

Where it's used

Spreadsheets and scientific calculators are full of rearranged formulas: loan terms, dosage calculations, unit conversions.

Compound InterestAlgebra · Economics & Finance

Applications of Linear Equations Quadratic Equations by Factoring and Square Roots