Changing the Subject of Formulae
Rearranging a formula, or changing the subject, is the process of isolating one variable on one side of the equals sign. This is a fundamental algebraic skill for manipulating equations to find a specific unknown quantity.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To isolate the target variable, apply inverse operations in the reverse order of operations (reverse BIDMAS/PEMDAS).
- Any operation (addition, multiplication, squaring, etc.) performed on one side of the equation must be performed on the other to maintain equality.
- If the target variable appears in multiple terms, collect all these terms on one side of the equation and factorise the variable out.
- Eliminate fractions early by multiplying every term on both sides by a common denominator.
- Remove square roots by squaring both sides of the equation. Similarly, remove powers by taking the appropriate root of both sides.
Formulae
If y = a(x+b), then x = (y/a) - b This is not a formula to memorise, but an example of the process. To make 'x' the subject, you undo the operations applied to it: first divide by 'a', then subtract 'b'.
Definitions
- Subject of a formula
- The variable that is isolated on one side of an equation, expressed in terms of the other variables and constants.
Worked example
Make g the subject of the formula: 5h = p(g - 1) - g/3
- 1
First, eliminate the fraction by multiplying every term on both sides by 3:
3 × 5h = 3 × p(g - 1) - 3 × (g/3), which simplifies to 15h = 3p(g - 1) - g - 2
Expand the bracket on the right side:
15h = 3pg - 3p - g - 3
Gather all terms containing 'g' onto one side and all other terms on the other.
Add 'g' and '3p' to both sides:
15h + 3p = 3pg - g - 4
Factorise 'g' out from the right-hand side:
15h + 3p = g(3p - 1) - 5
Isolate 'g' by dividing both sides by the bracket (3p - 1):
g = (15h + 3p) / (3p - 1)
Answer: g = (15h + 3p) / (3p - 1)
Common mistakes
- ×Sign errors are the most frequent mistake. When moving a term across the equals sign, be extremely careful to change its sign (e.g., `-g` becomes `+g`).
- ×When clearing a fraction, forgetting to multiply every single term by the denominator. For example, in `A = B + C/2`, multiplying by 2 gives `2A = 2B + C`, not `2A = B + C`.
- ×Errors in expanding brackets, especially with a negative sign outside, such as `-3(x - 2)` becoming `-3x - 6` instead of the correct `-3x + 6`.
No-calculator tips
- ✓Do not skip steps. Write down each transformation of the formula on a new line. Trying to perform multiple operations in your head increases the risk of sign or arithmetic errors.
- ✓Before starting, check if the subject variable appears more than once. If so, you know your strategy must involve collecting terms and factorising.
- ✓Once you have your final answer, do a quick mental substitution of a simple number (like 1 or 2) into the original and your rearranged formula to see if they are consistent. This can catch major errors.