Rearranging Complex Formulae

At GCSE Higher Tier, changing the subject of a formula becomes more demanding when the target variable appears more than once. When this happens, simple reverse operations are not enough. You must gather all target terms on one side, clean up brackets, and factorise to unlock the variable.

a) Factorising Multi-Term Subject Variables:

If the target variable appears in two separate terms on the same line, group them on one side of the equals sign and factorise it out into a single bracket.
Rearrange bx − 7 = 2x − w to make x the subject.
Step 1: Get all x terms on one side  →  bx − 2x − 7 = −w
Step 2: Move non-x terms to the other side  →  bx − 2x = 7 − w
Step 3: Factorise out the x variable  →  x(b − 2) = 7 − w
Step 4: Divide by the bracket  →  x = (7 − w) ÷ (b − 2)


b) Clearing Fractional Algebraic Denominators:

When your target variable is trapped inside the denominator of an algebraic fraction, multiply both sides by the entire denominator expression first.
Rearrange p = (x + 7) ÷ (3x − 2) to make x the subject.
Step 1: Multiply out the fraction base  →  p(3x − 2) = x + 7
Step 2: Expand the left bracket  →  3px − 2p = x + 7
Step 3: Group the x terms together  →  3px − x = 2p + 7
Step 4: Factorise out the target x  →  x(3p − 1) = 2p + 7
Step 5: Isolate x completely  →  x = (2p + 7) ÷ (3p − 1)


c) Dealing with Embedded Fractions:

If the formula has a lonely term next to a fraction, isolate the fractional chunk first before performing your cross-multiplication steps.
Rearrange w = 5 − 4 ÷ (2x + 3) to make x the subject.
Step 1: Isolate the fraction segment  →  4 ÷ (2x + 3) = 5 − w
Step 2: Multiply by the bracket base  →  4 = (5 − w)(2x + 3)
Step 3: Divide by the non-x bracket  →  4 ÷ (5 − w) = 2x + 3
Step 4: Complete standard linear steps  →  2x = [4 ÷ (5 − w)] − 3  →  x = ([4 ÷ (5 − w)] − 3) ÷ 2


The rules

1. Do Not Forget the Invisible Coefficient of 1
   When factorising terms like 3px − x, remember that the single x term has an implied coefficient of 1. Factorising yields x(3p − 1), not x(3p).
   
   
2. Expand Brackets to Set Variables Free
   If your target variable is locked inside a bracket that is being multiplied, you must expand that bracket completely before you can regroup your terms.
   
   
3. Keep Negative Signs with their Terms
   Be extra vigilant with subtraction signs when rearranging. Moving a negative term across the equals sign flips it into a positive addition term.
   
   
4. Factorising is the Master Unlock Step
   Whenever a variable appears twice and cannot be simplified further, factorisation is always the required tool to isolate it into a single position.
   

---

Difficulty levels:

This is a table of set questions that do not change, and as such as with all other lists in this system the questions are unaffected by the difficulty level selected.