Chapter 1 Linear Equations

# 1.2 Solve a Formula for a Specific Variable

Learning Objective

By the end of this section, you will be able to:

• Solve a formula for a specific variable

# Solve a Formula for a Specific Variable

We have all probably worked with formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.

It is often helpful to solve a formula for a specific variable. We call these formulas literal equations. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one, and all other variables and constants are on the other side of the equal sign.

Example 1

Solve the formula $A=P+Prt$ for t.

$\begin{array}{lrcl} \text{We will isolate t on one}\\ \text{side of the equation.} & A & = & P+Prt\\ \text{Subtract P from both sides to}\\ \text{isolate the term with t.} & A{-P} & = & P{-P}+Prt \\ \text{Simplify.} & A-P & = & Prt\\ \text{Divide both sides by Pr.} & \frac{A-P}{{Pr}} & = & \frac{Prt}{{Pr}}\\ \text{Simplify.} & t & = & \frac{A-P}{Pr}\\ \end{array}$

Exercise 1

Solve the formula $A=P+Prt$ for r.

Solution

$r=\frac{A-P}{Pt}$

Solve the formula $I=Prt$ for t.

Solution

$t=\frac{I}{Pr}$

Sometimes we might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

Example 2

Solve the formula $8x+7y=15$ for y.

$\begin{array}{lrcl} \text{We will isolate y on one side of}\\ \text{the equation.} & 8x+7y & = & 15\\ \text{Subtract 8x from both sides to}\\ \text{isolate the term with y.} & 8x{-8x}+7y & = & 15{-8x}\\ \text{Simplify.} & 7y & = & 15-8x\\ \text{Divide both sides by 7.} & \frac{7y}{{7}} & = & \frac{15-8x}{{7}}\\ \text{Simplify.} & y & = & \frac{15-8x}{7}\\ \end{array}$

Exercise 2

Solve the formula $4x+7y=9$ for y.

Solution

$y=\frac{9-4x}{7}$

Solve the formula $5x+8y=1$ for y.

Solution

$y=\frac{1-5x}{8}$

Access this online resource for additional instruction and practice with solving for a variable in literal equations.

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