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All courses Math II · A-CED.4 4 of 73
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.

Rearrange a formula to solve for a chosen variable

Problem
Rearrange this equation to solve for x: y=mx+b
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Hint

Undo the \(+b\) first so the term with \(x\) is by itself. Start from \(y=mx+b\) and subtract \(b\) from both sides.

To solve for \(x\), isolate the term \(mx\) first, then divide both sides by \(m\). Dividing by \(m\) requires \(m\ne 0\).

Solution walkthrough

01

Start with the original equation

\[y~=~\text{mx}+b\]

The goal is to isolate \(x\) on one side of the equation.

02

Subtract b from both sides

\[y-b~=~\text{mx}\]

Subtracting \(b\) undoes the constant term and leaves the \(mx\) term by itself.

03

Divide both sides by m

\[(y-b)/m~=~x,~m~\ne~0\]

Dividing by \(m\) isolates \(x\) because \(mx\) means \(m\) times \(x\). This division is valid only when \(m\ne 0\).

04

Write the rearranged equation

\[x~=~(y-b)/m,~m~\ne~0\]

This is the equation solved for \(x\), with the needed condition on \(m\).

+

Another way

  1. Check by substituting \(x=(y-b)/m\) into \(y=mx+b\): \(m((y-b)/m)+b=y-b+b=y\), so the rearranged formula returns the original equation when \(m\ne 0\).

!

Common mistake

A common mistake is to divide by \(m\) before subtracting \(b\), which incorrectly gives \(x=y/m+b\). The constant \(b\) must be moved before dividing the entire remaining expression by \(m\).