Rearranging formulae

In a nutshell

Rearranging or changing the subject means moving the terms in an equation around. The aim is usually to get a variable in a formula by itself on one side of the equation, with all other terms and numbers on the other side of the equation. The equation or formula is the same, it is just displayed in a different way. Rearranging for a particular variable makes it easier to calculate its value.

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Rearrange a formula

To rearrange an equation for a particular variable, you must take the other variables over to the other side of the equation. This is so that the variable you want to calculate is by itself on one side of the equation, and all other variables or numbers are on the other side. Whatever you do to one side of the equation, do the same to the other. In the formula

​$$x+y=z$$

to rearrange for $$x$$, decide which variables to move to the other side, in this case $$y$$. Think about what operation $$y$$ is doing, here it is $$+y$$. The formula has $$+y$$, so do the inverse operation and $$-y$$ to both sides of the equation. ​

​$$\begin {aligned}\qquad x+y&=z \\-y \qquad & \qquad -y\\ \end {aligned}\\ \qquad \underline{x = z-y}$$

Inverse operations

Inverse operations perform the opposite function. For example, addition is opposite to subtraction, or multiplication is opposite to division.

Example 1

​$$F=ma$$

Rearrange for $$a$$ and calculate $$a$$ if $$F=250$$ and $$m=80$$

Answer

First rearrange for $$a$$​

​$$\begin {aligned}\qquad F&=ma \\\div m \qquad& \qquad \div m \\\qquad \frac F m &=a \\\qquad a&= \frac F m \\\end {aligned}$$​​

Now substitute numbers

$$\qquad a= \frac {250} {80} = \underline{3.125}$$​​​

Example 2

Rearrange $$v=u+at$$ for $$t$$

Answer

$$\begin {aligned}v&=u+at \\-u \qquad &\qquad -u \\v-u &= at \\\div a \qquad &\qquad \div a \\\frac {v-u} a &= t \\\end {aligned}\\ \qquad \underline{t= \frac {v-u} a}$$