Lesson 4: Rewriting Equations and Formulas

Property

To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all the other variables and constants on the other side. This process is also called solving a literal equation.

Examples

• To solve the perimeter formula for a rectangle, $P = 2L + 2W$, for $L$, first subtract $2W$ to get $P - 2W = 2L$. Then divide by 2: $L = \frac{P - 2W}{2}$.

• To solve the simple interest formula $I = Prt$ for the principal $P$, you divide both sides by the product of rate and time: $P = \frac{I}{rt}$.

Property

To solve a formula for a specific variable when fractions are present, first clear the fractions by multiplying the entire equation by the denominator or LCD. Then, use inverse operations to isolate the desired variable on one side of the equation.

Examples

Property

When solving a formula that includes a fraction, like the area of a triangle $A = \frac{1}{2}bh$, you can first clear the fraction by multiplying both sides of the equation by the denominator.
Then, isolate the desired variable. To solve for $h$, we get $2A = bh$, which simplifies to $h = \frac{2A}{b}$.

Examples

• Solve the formula $A = \frac{1}{2}bh$ for $b$. First, multiply by 2 to get $2A = bh$. Then, divide by $h$ to get $b = \frac{2A}{h}$.
• The formula for the average of two numbers is $M = \frac{a+b}{2}$. Solve for $a$. Multiply by 2 to get $2M = a+b$. Then subtract $b$ to get $a = 2M - b$.
• Solve the formula for the area of a trapezoid, $A = \frac{1}{2}h(b_1 + b_2)$, for $h$. Multiply by 2: $2A = h(b_1 + b_2)$. Then divide by $(b_1+b_2)$ to get $h = \frac{2A}{b_1 + b_2}$.

Explanation

To get a variable out of a fraction, first 'free' it by multiplying everything by the denominator. This clears the fraction and makes it much easier to isolate the variable you're looking for using standard algebra steps.