Formulae & Substitution

Spec reference: Algebra - Notation, vocabulary and manipulation
Key idea: Substitute values into formulae and rearrange formulae to change the subject.

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Substitution

Replace letters with given values and evaluate.

Example

Given $v=u+at$, find $v$ when $u=5$, $a=3$, $t=4$.

$v=5+(3)(4)=5+12=\boxed{{\displaystyle 17}}$

Example

Given $E=\frac{1}{2}m{v}^2$, find $E$ when $m=4$ and $v=6$.

$E=\frac{1}{2}\times 4\times 36=\boxed{{\displaystyle 72}}$

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Changing the subject

Use inverse operations to make a different variable the subject.

Example

Make $u$ the subject of $v=u+at$

$u=v-at$

Example

Make $r$ the subject of $A=\pi r^2$

$r^2={\displaystyle \frac{A}{\pi }}$

$r=\sqrt{{\displaystyle \frac{A}{\pi }}}$

Example

Make $x$ the subject of $y={\displaystyle \frac{3x+1}{x-2}}$

$y(x-2)=3x+1$

$xy-2y=3x+1$

$xy-3x=2y+1$

$x(y-3)=2y+1$

$x={\displaystyle \frac{2y+1}{y-3}}$

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Common formulae to know

Formula	Meaning
$v=u+at$	Velocity (kinematics)
$s=\frac{1}{2}(u+v)t$	Distance (kinematics)
$A=\pi r^2$	Area of circle
$C=2\pi r$	Circumference
$V=lwh$	Volume of cuboid
$A=\frac{1}{2}bh$	Area of triangle

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Exam tips

Watch out for

• Substitute before simplifying - avoid making arithmetic mistakes in your head
• When changing the subject, if the subject appears more than once, factorise it out
• Always show each step clearly

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Test Yourself

Question 1 of 5

Given $v = u + at$ where $u=3$, $a=2$, $t=5$ Find $v$.