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Rearranging formulae

Rearranging formulae

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Explainer Video

Tutor: Meera

Summary

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=zx+y=zx+y=z


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


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



Inverse operations

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

​+++​​
​−-−​​
​×\times×​​
​÷\div÷​​
​x2x^2x2​​
​x\sqrt{x}x​​​


Example 1

​F=maF=maF=ma

Rearrange for aaa and calculate aaa if F=250F=250F=250 and m=80m=80m=80


First rearrange for aaa. To get aaa by itself, move mmm to the other side. As mmm is multiplying aaa, perform the inverse operation and divide by mmm to both sides.​

​F=ma÷m÷mFm=aa=Fm‾\begin {aligned}\qquad F&=ma \\\div m \qquad& \qquad \div m \\\qquad \frac F m &=a \\ \end {aligned}\\\quad \underline{a= \frac F m}F÷mmF​​=ma÷m=a​a=mF​​​​


Now substitute numbers

a=25080=3.125‾\qquad a= \frac {250} {80} = \underline{3.125}a=80250​=3.125​


Example 2

Rearrange v=u+atv=u+atv=u+at for ttt


To get ttt by itself, on one side of the equation, you need to move uuu and aaa. 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 ttt, you would multiply by aaa first, then add uuu. So when rearranging, reverse this process and move uuu first, then aaa. ​

​v=u+at−u−uv−u=at÷a÷av−ua=tt=v−ua‾\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 }v−uv−u÷aav−u​​=u+at−u=at÷a=t​t=av−u​​​​


Example 3

Rearrange y=x+1x−2y = \frac {x+1} {x-2}y=x−2x+1​ for xxx​​


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.

​y=x+1x−2×(x−2)y(x−2)=x+1xy−2y=x+1−xxy−2y−x=1+2yxy−x=1+2yx(y−1)=1+2yx=1+2yy−1‾\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}yy(x−2)xy−2yxy−2y−xxy−xx(y−1)x​=x−2x+1​=x+1=x+1=1=1+2y=1+2y=y−11+2y​​​×(x−2)−x+2y​



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Rearranging formulae

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FAQs - Frequently Asked Questions

What does rearrange mean in maths?

Rearranging means we are moving the terms in an equation around to help us find the answer. It is a different way of displaying an equation.

What are inverse operations?

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

Why do we rearrange in algebra?

We rearrange an equation to display the equation in a different way. We rearrange equations and formulae to make one variable the subject, to make it easier to calculate its value.

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