Can I rearrange any formula for any variable?

Yes, you can rearrange any equation to solve for any variable in it. However, the rearranged expression may be more or less complex. For simple linear formulas like area, the result is straightforward; for more complicated relationships, you may end up with fractions or multiple variables on one side.

Do I have to rearrange, or can I just plug in numbers?

You can substitute numbers directly into the original formula, but rearranging first gives you a formula you can reuse for many different values. This saves time if you solve the same problem multiple times with different numbers.

What if my rearranged formula has multiple variables on one side?

That is normal and correct. For example, solving A = \ell w for \ell gives \ell = \frac{A}{w}, which has A and w on the right side. This is exactly what you want—it tells you how to compute \ell when you know the other two values.

How do I know which operation to use when rearranging?

Look at the original formula and identify what operation connects the variable you want to the rest. If the variable is multiplied by something, divide to undo it. If it is added to something, subtract to undo it. Always apply the inverse (opposite) operation to both sides.

Literal Equations

Literal equations rearrange formulas to solve for any variable, letting you find missing measurements when you know the others.

A = \ell w \implies w = \frac{A}{\ell}

Solve a problem with Literal Equations

Type the problem. The solver will use Literal Equations where Literal Equations is the right tool, and tell you when it is not.

What each symbol means

What Literal Equations takes

Symbol	Meaning
$A$	The area of a rectangle, measured in square units; treating it as just a number instead of a variable that depends on length and width breaks the ability to rearrange the formula.
$\ell$	The length (pronounced 'ell'), one dimension of a rectangle; it looks like the letter l or the number 1, so use the LaTeX symbol to avoid confusion in written work.
$w$	The width, the other dimension; forgetting that $w$ is a variable you can solve for, rather than a fixed constant, means you cannot isolate it on one side of the equation.

When to use it

Use this when you have a formula but need to solve for a different variable than usual.

Level

Usually taught in: Algebra I

Worked examples

1. Solve for width from an area formula

Problem

Solve the formula

A = \ell w

for

w

$A = \ell w$
This is the given formula for the area of a rectangle.
$\frac{A}{\ell} = w$
Divide both sides by $\ell$ to undo the multiplication and isolate $w$ .
$w = \frac{A}{\ell}$
Rewrite with $w$ on the left side for standard form.

Answer: $w = \frac{A}{\ell}$

To solve for $w$ , we undid the multiplication by $\ell$ using division. This rearranged formula lets you find width when you know area and length. The method works for any values of $A$ and $\ell$ .

2. Solve for cost when profit is known

Problem

The profit formula is

P = R - C

, where

P

is profit,

R

is revenue, and

C

is cost. Solve for

C

$P = R - C$
This is the given profit formula.
$P - R = -C$
Subtract $R$ from both sides to isolate the term containing $C$ .
$C = R - P$
Divide both sides by $-1$ (or multiply by $-1$ ) to make $C$ positive.

Answer: $C = R - P$

Subtraction requires tracking signs more carefully than multiplication does. Once we isolate $-C$ , we divide by the coefficient $-1$ to get $C$ by itself. This formula shows that cost equals revenue minus profit.

3. Find a monthly fee from a phone bill

Problem

A phone plan charges a monthly base fee of

F

dollars, plus $0.50 per gigabyte used. Your bill is

B = F + 0.50g

, where

g

is gigabytes. If your bill was $65 for 10 GB of usage, solve for the monthly fee.

$B = F + 0.50g$
This is the given formula for the total bill.
$F = B - 0.50g$
Subtract $0.50g$ from both sides to isolate the monthly fee $F$ .
$F = 65 - 0.50(10)$
Substitute $B = 65$ and $g = 10$ into the rearranged formula.
$F = 65 - 5 = 60$
Calculate $0.50 \times 10 = 5$ , then subtract to get $F = 60$ .

Answer: $F = 60$

We first rearranged the formula to isolate $F$ , then substituted the values $B = 65$ and $g = 10$ from the problem. The monthly fee was 60 dollars. By rearranging first, we created a formula we could use for any bill amount.

Common mistakes

Where Literal Equations usually goes wrong

The mistake	Why it is wrong	The fix
Solving $A = \ell w$ for $w$ gives $w = \frac{\ell}{A}$ (reversing numerator and denominator).	Students reverse the numerator and denominator when dividing, putting the multiplier on top instead of the product.	In the original formula, $\ell$ multiplies $w$ , so divide the product $A$ by the multiplier $\ell$ : $w = \frac{A}{\ell}$ .
Solving $A = \ell w$ for $w$ by writing $w = A - \ell$ (subtracting instead of dividing).	Students confuse subtraction with division when trying to isolate a variable, especially if they are not carefully tracking what operation connects the variables.	Ask: what operation connects $A$ , $\ell$ , and $w$ ? It is multiplication ( $\ell \times w = A$ ), so undo it with division: $w = \frac{A}{\ell}$ .
Solving $B = F + 0.50g$ for $F$ , but writing $F = 0.50g - B$ (reversing the subtraction).	When subtracting the same quantity from both sides, students sometimes reverse the order of the terms on the right side.	Subtract $0.50g$ from both sides carefully: $B - 0.50g = F + 0.50g - 0.50g$ gives $F = B - 0.50g$ (not $0.50g - B$ ).

Tips and when to use something else

Always perform the same operation on both sides of the equation to keep it balanced.
Identify the operation connecting the variable you want to the rest of the formula, then apply its inverse: undo addition with subtraction, undo multiplication with division.
Check your rearrangement by substituting simple test numbers into both the original and rearranged formulas—they should give the same result.
Once you can rearrange one formula, try Solving by Substitution when you have two equations and need to find the values of two unknowns.

Related formulas

Linear EquationSolve for the value of x in any linear equation by rearranging to isolate x, finding the unique solution where the expression equals zero.Two-Step EquationsSolve two-step equations of the form ax + b = c by undoing operations in reverse order to isolate the variable using inverse operations.Solving by SubstitutionSolving by Substitution solves a system of equations by replacing one variable with its equivalent expression, turning two equations into one.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.

Frequently asked questions

Can I rearrange any formula for any variable?: Yes, you can rearrange any equation to solve for any variable in it. However, the rearranged expression may be more or less complex. For simple linear formulas like area, the result is straightforward; for more complicated relationships, you may end up with fractions or multiple variables on one side.
Do I have to rearrange, or can I just plug in numbers?: You can substitute numbers directly into the original formula, but rearranging first gives you a formula you can reuse for many different values. This saves time if you solve the same problem multiple times with different numbers.
What if my rearranged formula has multiple variables on one side?: That is normal and correct. For example, solving $A = \ell w$ for $\ell$ gives $\ell = \frac{A}{w}$ , which has $A$ and $w$ on the right side. This is exactly what you want—it tells you how to compute $\ell$ when you know the other two values.
How do I know which operation to use when rearranging?: Look at the original formula and identify what operation connects the variable you want to the rest. If the variable is multiplied by something, divide to undo it. If it is added to something, subtract to undo it. Always apply the inverse (opposite) operation to both sides.

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18