Lesson 4: Rewriting 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

The formula $I = Prt$ calculates simple interest, where $I$ is interest, $P$ is principal, $r$ is rate, and $t$ is time. To solve for a different variable, such as the principal ($P$), you isolate it by dividing both sides by the other variables. The formula for principal becomes $P = \frac{I}{rt}$.

Examples

• Solve $I=Prt$ for $t$. To isolate time, divide the interest by the principal and rate. The formula is $t = \frac{I}{Pr}$.
• Find the principal $P$ if the interest earned $I$ was 300 dollars, the rate $r$ was $5\% (0.05)$, and the time $t$ was 2 years. Using $P = \frac{I}{rt}$, we get $P = \frac{300}{0.05 \cdot 2} = \frac{300}{0.1} = 3000$ dollars.
• Find the rate $r$ if 10000 dollars principal ($P$) earned 2400 dollars interest ($I$) over 3 years ($t$). Using $r = \frac{I}{Pt}$, we get $r = \frac{2400}{10000 \cdot 3} = \frac{2400}{30000} = 0.08$ or $8\%$.

Explanation

By rearranging the interest formula, you can find the initial investment (principal), the interest rate, or the time. This is useful for financial planning when you know your goal but need to figure out one of the starting conditions.