Lesson 4: Literal Equations and Formulas

Property

A formula is an equation that states a relationship between two or more quantities, often used in a specific context like geometry or physics.
A literal equation is an equation that involves two or more variables. All formulas are literal equations.

Examples

• The formula for the perimeter of a rectangle, $P = 2l + 2w$, is a literal equation.
• The standard form of a linear equation, $Ax + By = C$, is a literal equation.
• The formula for simple interest, $I = prt$, is a literal equation relating interest, principal, rate, and time.

Explanation

Literal equations and formulas are fundamental in mathematics and science because they express general relationships. Instead of using specific numbers, they use variables to represent quantities. By understanding that these are equations, you can apply the same properties of equality to rearrange them and solve for any one of the variables in terms of the others.

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

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.