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Formulae: Changing the Subject
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Foundational Skills
Identify the operation applied to a variable
\[ y = 5x \rightarrow \text{multiply by 5} \]
Identify whether a variable is being added to, subtracted from, multiplied, or divided.
State the inverse operation
\[ \text{add 5} \rightarrow \text{subtract 5} \]
State the inverse of a given arithmetic operation.
Identify the order of operations to undo
\[ y = 3x + 2 \rightarrow \text{subtract first} \]
Identify the correct order for applying inverse operations.
One-step Rearranging
Rearrange by adding or subtracting
\[ y = x + 5 \rightarrow x = y – 5 \]
Rearrange a formula by performing a single addition or subtraction.
Rearrange by multiplying or dividing
\[ y = 3x \rightarrow x = \frac{y}{3} \]
Rearrange a formula by performing a single multiplication or division.
Rearrange when subject is subtracted from a constant
\[ y = a – x \rightarrow x = a – y \]
Rearrange where the subject is subtracted from a constant.
Two-step Rearranging
Rearrange requiring two inverse operations
\[ y = 2x + 3 \rightarrow x = \frac{y-3}{2} \]
Apply two inverse operations in the correct order.
Rearrange with subject on denominator
\[ s = \frac{d}{t} \rightarrow t = \frac{d}{s} \]
Rearrange where the subject is on the denominator.
Rearrange with a negative coefficient
\[ y = a – bx \rightarrow x = \frac{a-y}{b} \]
Rearrange where the subject has a negative coefficient.
Subject multiplied by multiple terms
\[ A = 2\pi rh \rightarrow r = \frac{A}{2\pi h} \]
Rearrange where the subject is multiplied by multiple terms.
Rearranging with Powers
Rearrange when the subject is squared
\[ A = \pi r^2 \rightarrow r = \sqrt{\frac{A}{\pi}} \]
Rearrange where the subject is squared.
Rearrange when subject is under a square root
\[ y = \sqrt{x} \rightarrow x = y^2 \]
Rearrange where the subject is inside a square root.
Rearrange with cubes or cube roots
\[ V = s^3 \rightarrow s = \sqrt[3]{V} \]
Rearrange where the subject is cubed or under a cube root.
Rearrange with higher powers or roots
\[ y = x^4 \rightarrow x = \sqrt[4]{y} \]
Rearrange where the subject has powers other than 2 or 3.
Rearranging with Brackets
Rearrange formulae with brackets
\[ y = a(x+b) \rightarrow x = \frac{y}{a} – b \]
Rearrange a formula containing brackets.
Rearrange with a fraction containing a sum
\[ y = \frac{x+a}{b} \rightarrow x = by – a \]
Subject is in the numerator with other terms.
Rearrange with brackets and powers
\[ y = (x+a)^2 \rightarrow x = \sqrt{y} – a \]
A bracket containing the subject is raised to a power.
Rearrange where answer contains a bracket
\[ y = \frac{a}{x+b} \rightarrow x = \frac{a}{y} – b \]
The final answer naturally contains a bracket.
Multi-step Rearranging
Rearrange complex formulae with multiple steps
\[ v^2 = u^2 + 2as \rightarrow u = \sqrt{v^2 – 2as} \]
Multiple steps including dealing with powers.
Rearrange involving reciprocals
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \rightarrow u = \frac{fv}{v-f} \]
Rearrange formulae involving reciprocals.
Rearrange with roots and fractions combined
\[ T = 2\pi\sqrt{\frac{l}{g}} \rightarrow l = \frac{gT^2}{4\pi^2} \]
Combining roots and fractions.
Extensions
Rearrange when subject appears twice
\[ y = \frac{x+a}{x+b} \rightarrow x = \frac{by-a}{1-y} \]
Subject appears more than once (requires factorising).
Rearrange with x in multiple linear terms
\[ ax + b = cx + d \rightarrow x = \frac{d-b}{a-c} \]
Subject appears in two linear terms.
Rearrange in context and interpret
\[ C = \frac{5(F-32)}{9} \rightarrow F = \frac{9C}{5} + 32 \]
Rearrange in a real-world context.
Identify the correct rearrangement
\[ \text{Which is correct?} \]
Identify which of two rearrangements is correct.
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