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All courses Math III · A-CED.4 11 of 55
Rearrange formulas to highlight a chosen quantity across the expression types studied.

Rearrange a formula to isolate a linear target variable in one term

Problem
Rearrange this formula to isolate the linear target variable: A=lw isolate l
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Hint

Divide both sides by the factor attached to \(l\).

In a product formula like \(A=lw\), isolate one variable by dividing by the other factor, as long as that factor is not \(0\).

Solution walkthrough

01

Start with the formula

\[A~=~\text{lw}\]

The variable \(l\) is multiplied by \(w\), so we need to undo that multiplication.

02

Divide both sides by `w`

\[A/w~=~(\text{lw})/w\]

Dividing both sides by \(w\) isolates \(l\), provided \(w\) is not \(0\).

03

Simplify the right side

\[l~=~A/w\]

The factor \(w\) cancels on the right, leaving \(l\) by itself.

04

State the restriction

\[w~\ne~0\]

Division by \(0\) is undefined, so this rearrangement requires \(w\) to be nonzero.

05

State the final answer

\[A~\text{correct}~\text{rearrangement}~\text{is}~l~=~A/w,~\text{with}~w~\text{not}~\text{equal}~0.\]

A correct rearrangement is \(l=A/w\), with \(w not equal 0\).

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Another way

  1. You can think of \(w\) as the multiplicative coefficient of \(l\) and undo it in one step by dividing both sides by \(w\).

!

Common mistake

A common mistake is to write \(l=A-w\), even though \(l\) is multiplied by \(w\), not added to it.