1-5 Using Formulas

Property

When the variable you want to isolate appears in more than one term, factor it out first, then divide both sides by the remaining expression.

If $p$ appears in two terms, rewrite as:

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.

Property

Dimensional analysis is a method to convert units by multiplying the original quantity by a series of unit fractions.
A unit fraction is a ratio of equivalent measurements that equals 1 (e.g., $\frac{1 \ \operatorname{foot}}{12 \ \operatorname{inches}} = 1$).
This process cancels out unwanted units until only the desired units remain. This method relies on two key properties:

• For any nonzero number $n$, $\frac{n}{n} = 1$.
• Any number $n$ multiplied by 1 is still $n$: $n \cdot 1 = n$.

Examples

• To convert 7000 pounds to tons, use the unit fraction relating tons and pounds: $7000 \ \operatorname{lbs} \times \frac{1 \ \operatorname{ton}}{2000 \ \operatorname{lbs}} = \frac{7000}{2000} \ \operatorname{tons} = 3.5 \ \operatorname{tons}$.
• To convert 2.5 hours into seconds, use a chain of unit fractions: $2.5 \ \operatorname{hours} \times \frac{60 \ \operatorname{minutes}}{1 \ \operatorname{hour}} \times \frac{60 \ \operatorname{seconds}}{1 \ \operatorname{minute}} = 2.5 \times 60 \times 60 \ \operatorname{seconds} = 9000 \ \operatorname{seconds}$.
• How many kilograms are in 55 pounds? Use the conversion 1 lb = 0.454 kg: $55 \ \operatorname{lbs} \times \frac{0.454 \ \operatorname{kg}}{1 \ \operatorname{lb}} = 55 \times 0.454 \ \operatorname{kg} = 24.97 \ \operatorname{kg}$.

Explanation

This is a powerful technique for converting any unit. You create a chain of fractions where the unit names you want to eliminate cancel out from the numerator and denominator, leaving you with only the units you need for your answer.