Literal Equations and Formulas: Practice

The unknown is the letter you are asked to isolate. Here that is $x$.

$\text{solve } ax = b \text{ for } x \;\Rightarrow\; x \text{ is the unknown}$

Choosing $x$ as the thing to solve for makes $a$ and $b$ the parameters.

A letter held fixed while you isolate another is a parameter, a stand-in for a specific but unstated number.

$a, b \text{ held fixed}, \qquad x \text{ isolated}$

So $a$ and $b$ are parameters and $x$ is the unknown.

Divide both sides by the nonzero coefficient $a$.

$ax = b \;\Rightarrow\; x = \frac{b}{a}$

This step is valid precisely because $a \neq 0$.

With $a = 0$ the equation reads $0 \cdot x = 5$.

$0 \cdot x = 5 \;\Rightarrow\; 0 = 5$

The left side is $0$ for every $x$, never $5$, so there is no solution.

Here $a = 5 \neq 0$, so divide.

$x = \frac{b}{a} = \frac{20}{5} = 4$

Exactly one solution, so the solution set is $\{4\}$.

Divide both sides by $l$.

$A = lw \;\Rightarrow\; w = \frac{A}{l}$

The step is valid because $l \neq 0$.

Divide both sides by the coefficient $k$.

$y = kx \;\Rightarrow\; x = \frac{y}{k}$

Valid because $k \neq 0$.

Now $a$ is the unknown and $x$ is a parameter, so divide both sides by $x$.

$ax = b \;\Rightarrow\; a = \frac{b}{x}$

Same three symbols, but a different letter chosen to isolate, so the division is by $x$ and the condition is $x \neq 0$.

Divide both sides by the constant $4$, which is nonzero.

$P = 4s \;\Rightarrow\; s = \frac{P}{4}$

Divide both sides by $I$.

$V = IR \;\Rightarrow\; R = \frac{V}{I}$

Valid since $I \neq 0$.

A literal equation contains more than one letter.

$P = 2l + 2w \text{ has the letters } P, l, w$

The other three each contain only the single letter $x$.

Solving for a variable means isolating it, with every other letter gathered on the other side.

$d = rt \;\Rightarrow\; r = \frac{d}{t}$

So "solve for $r$" means rewrite the formula with $r$ alone on one side.