Learn on PengiBig Ideas Math, Algebra 1Chapter 1: Solving Linear Equations

Lesson 5: Rewriting Equations and Formulas

Property To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. For example, to solve $d = rt$ for $t$, we divide both sides by $r$ to get $t = \frac{d}{r}$.

Section 1

Solving a Formula for a Variable

Property

To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. For example, to solve $d = rt$ for $t$ , we divide both sides by $r$ to get $t = \frac{d}{r}$ .

Examples

Solve the formula $d=rt$ for $r$ . To isolate $r$ , we divide both sides by $t$ . The new formula is $r = \frac{d}{t}$ .
Solve the formula for the perimeter of a triangle, $P = a + b + c$ , for side $b$ . We subtract $a$ and $c$ from both sides, so $b = P - a - c$ .
Solve the formula for volume, $V = LWH$ , for the width $W$ . We divide both sides by $L$ and $H$ . The result is $W = \frac{V}{LH}$ .

Explanation

Rearranging a formula lets you find any one piece of information if you know the others. It’s like rewriting a recipe to figure out how much flour you need based on the number of cookies you want to bake.

Section 2

Solving for one variable

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

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solving a Formula for a Variable

Property

Examples

Solve the formula $d=rt$ for $r$ . To isolate $r$ , we divide both sides by $t$ . The new formula is $r = \frac{d}{t}$ .
Solve the formula for the perimeter of a triangle, $P = a + b + c$ , for side $b$ . We subtract $a$ and $c$ from both sides, so $b = P - a - c$ .
Solve the formula for volume, $V = LWH$ , for the width $W$ . We divide both sides by $L$ and $H$ . The result is $W = \frac{V}{LH}$ .

Explanation

Section 2

Solving for one variable

Property

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

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Solving a Formula for a Variable

Property

Examples

Solve the formula $d=rt$ for $r$ . To isolate $r$ , we divide both sides by $t$ . The new formula is $r = \frac{d}{t}$ .
Solve the formula for the perimeter of a triangle, $P = a + b + c$ , for side $b$ . We subtract $a$ and $c$ from both sides, so $b = P - a - c$ .
Solve the formula for volume, $V = LWH$ , for the width $W$ . We divide both sides by $L$ and $H$ . The result is $W = \frac{V}{LH}$ .

Explanation

Section 2

Solving for one variable

Property

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