# 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.

## 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.

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

## Inverse operations

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

$+$ | $-$ |

$\times$ | $\div$ |

$x^2$ | $\sqrt{x}$ |

##### Example 1

$F=ma$

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

*First rearrange for* $a$. *To get* $a$ *by itself, move* $m$ *to the other side. As* $m$ *is multiplying* $a$, *perform the inverse operation and divide by* $m$ *to both sides*.

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

*Now substitute numbers*

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

##### Example 2

*Rearrange* $v=u+at$ *for* $t$

*To get* $t$ *by itself, on one side of the equation, you need to move* $u$ *and* $a$. *To work out which variable to move first, think about what order you would perform a calculation, if you were given numbers to substitute in. If you were given a value for* $t$, *you would multiply by* $a$ *first, then add* $u$. *So when rearranging, reverse this process and move* $u$ *first, then* $a$.

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

##### Example 3

*Rearrange* $y = \frac {x+1} {x-2}$ *for* $x$

*When there are fractions, it is always best to multiply the formula up by the denoinator, to eliminate fractions first, then multiply out and rearrange.*

$\begin {aligned}y &= \frac {x+1} {x-2} \\&& \times (x-2) \\y(x-2) &= x+1 \\xy -2y &= x+1 \\&& -x \\xy-2y-x &= 1 \\&& +2y \\xy-x &=1+2y \\x(y-1) &= 1+2y \\x &= \underline{\frac {1+2y}{y-1}} \\\end {aligned}$