Lesson 5: Equations

Property

To solve a formula for one variable, treat it as the unknown and all other variables as constants.
Isolate the desired variable by applying inverse operations to both sides of the equation.
Remember to follow the order of operations in reverse and do not combine unlike terms.

Examples

• To solve the perimeter formula $P = 2l + 2w$ for $l$, first subtract $2w$ from both sides: $P - 2w = 2l$. Then, divide by 2: $l = \frac{P - 2w}{2}$.

• To solve the interest formula $A = P + Prt$ for $r$, first subtract $P$: $A - P = Prt$. Then, divide by $Pt$ to isolate $r$: $r = \frac{A - P}{Pt}$.

Property

To solve a formula for a specific variable, first clear any fractions by multiplying the entire equation by the LCD. Next, gather all terms containing the desired variable on one side of the equation. If there are multiple terms with the variable, factor the variable out, and then divide both sides by the remaining factor to isolate it.

Examples

• Solve the formula $S = \frac{a}{1-r}$ for $r$. Multiply by $1-r$ to get
  $$S(1-r) = a$$
  . Distribute $S$ to get
  $$S - Sr = a$$
  . Then,
  $$-Sr = a - S$$
  , so
  $$r = \frac{a-S}{-S}$$
  or
  $$r = \frac{S-a}{S}$$
  .
• Solve $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ for $u$. The LCD is $fuv$. Multiply by the LCD to get
  $$uv = fv + fu$$
  . Move terms with $u$ to one side:
  $$uv - fu = fv$$
  . Factor out $u$:
  $$u(v-f) = fv$$
  . The solution is
  $$u = \frac{fv}{v-f}$$
  .
• Solve $h = \frac{2A}{b_1 + b_2}$ for $b_1$. Multiply by $b_1+b_2$ to get
  $$h(b_1+b_2) = 2A$$
  . Distribute $h$:
  $$hb_1 + hb_2 = 2A$$
  . Then
  $$hb_1 = 2A - hb_2$$
  , and
  $$b_1 = \frac{2A - hb_2}{h}$$
  .

Explanation

Rearranging formulas is like solving a puzzle. First, get rid of fractions. Then, herd all the pieces with your target variable to one side. If it appears in multiple terms, factor it out to isolate it.