Rearranging Formulae to Change the Subject

Changing the Subject of a Formula

Changing the subject of a formula is the process of expressing one variable in terms of all the other variables in the equation. To do this, you must systematically rearrange the formula using the standard rules of arithmetic and algebra to isolate the new subject. teaching this effectively requires following strict algebraic balance: any operation performed on one side of the equals sign must be performed on the other.

The Fundamental Rules for Rearranging

According to the official guide, there are five key operations you can use to transform a formula:

1. Add or subtract the same quantity or term from each side.
2. Multiply or divide both sides by the same non-zero quantity.
3. Invert both sides of a formula if both sides are single fractions with non-zero denominators.
4. Square both sides of a formula or raise both sides to the same non-zero power.
5. Take the square root of both sides of a formula.

Changing the Subject Using Subtraction

In many cases, the variable you wish to isolate appears in multiple terms. The goal is to gather all terms containing that variable on one side of the equation while moving all other terms to the opposite side.

Worked Example: Making $p$ the Subject

Consider the formula: $q + 2p = p + 4q - 6r$

1. Subtract $p$ from both sides to move all terms involving $p$ to the left: $q + 2p - p = 4q - 6r$.
2. Collect like terms: $q + p = 4q - 6r$.
3. Subtract $q$ from both sides to isolate the $p$ term: $p = 4q - 6r - q$.
4. Simplify by collecting the remaining like terms: $p = 3q - 6r$.
5. Factorise by taking out the common factor of 3 to provide a clean final form: $p = 3(q - 2r)$.

Rearranging with Multiple Functions

When a formula includes brackets and fractions, it is usually most efficient to expand the brackets and clear the denominators early in the process.

Worked Example: Making $x$ the Subject

Consider the formula: $y = 3(x - 2) + \frac{x}{2}$

1. Multiply out the bracket first: $y = 3x - 6 + \frac{x}{2}$.
2. Multiply both sides of the entire formula by 2 to eliminate the fraction: $2y = 6x - 12 + x$.
3. Collect the $x$ terms on the right-hand side: $2y = 7x - 12$.
4. Add 12 to both sides to further isolate the $x$ term: $2y + 12 = 7x$.
5. Finally, divide by 7 to make $x$ the subject: $x = \frac{2y + 12}{7}$.

Variables in the Denominator

If the required subject is part of a denominator, you must use multiplication or inversion to bring it to the numerator. This often requires combining terms into a single fraction first.

Worked Example: Making $u$ the Subject

Consider the formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{p}$

1. Rearrange the equation to isolate the term containing $u$: $\frac{1}{v} - \frac{1}{p} = \frac{1}{u}$.
2. To prepare for inversion, combine the left-hand side into a single fraction using a common denominator of $vp$: $\frac{p - v}{vp} = \frac{1}{u}$.
3. Invert both sides of the equation (turn the fractions upside down): $\frac{vp}{p - v} = \frac{u}{1} = u$.

Formulae Involving Square Roots

When the subject is trapped inside a square root, you must isolate the root and then square the entire side of the equation.

Worked Example: Making $x$ the Subject

Consider the formula: $y = \sqrt{\frac{3x + 5}{4}}$

1. Square both sides of the formula to remove the radical: $y^2 = \frac{3x + 5}{4}$.
2. Multiply both sides by 4: $4y^2 = 3x + 5$.
3. Subtract 5 from both sides: $4y^2 - 5 = 3x$.
4. Divide the whole expression by 3 to isolate $x$: $x = \frac{4y^2 - 5}{3}$.