Harder rearranging formulae | Mathematics

Unit 1 · Algebra: changing the subject

2 Harder Rearranging Formulae

A formula is just an equation wearing a different variable's name as the subject. Everything you already know about solving equations still works — the only new problem is that x can be hiding in three different ways: behind layers, on both sides at once, or buried inside a fraction.

🎯 By the end of this you can…

✓Make x the subject when it appears once, even behind a bracket, a fraction or a square.
✓Collect x onto one side when it appears twice, then factorise it out.
✓Clear a fraction with x in the denominator before doing anything else.

💡 The idea

Isolating x — same moves, more layers

When x appears exactly once, the method is the one you already know: undo whatever was done to x, in the reverse of the order it was built up — exactly like reverse BIDMAS. The only difference here is how many layers are stacked on x: a number added, then a division, then even a square. Peel them off one at a time, outermost first.

📋 How to undo the layers

1
Find the LAST thing that was done to x — that's the first thing you undo.
2
Undo it with the opposite operation, on both sides.
3
Repeat until x is completely alone.
4
If the last layer was a square, undo it with a square root — and remember $\pm$ .

✏️ Worked example

Make x the subject: $P = \frac{x ^{2}}{4} + c$

1
Subtract c from both sides — it's the last thing added: $P - c = \frac{x ^{2}}{4}$
2
Multiply both sides by 4, to undo the division: $x^{2} = 4 (P - c)$
3
Square root both sides — the LAST thing left is the square, so it's the FIRST thing undone here, and a square root always needs both signs: $x = \pm 2 P - c$

✅ Answer: $x = \pm 2 P - c$

⚠️ The classic trap

Forgetting the ± the moment a square root appears. $x^{2} = 16$ has TWO solutions, $x = 4$ and $x = - 4$ — a formula is no different. Unless a question specifically asks for "the positive value" (like the practice below does, to keep the numbers simple), the subject of a formula built from $x^{2}$ keeps its ±.

💡 The idea

x on both sides — bring it home first

The moment x appears on BOTH sides, "undo the last operation" doesn't work any more — there isn't one x to undo, there are two. So the very first move changes completely: before you divide or subtract anything else, drag every x-term onto the same side, and every non-x term onto the other. Only once every x is together — even if it's wearing two different coats, like $a x$ and $b x$ — can you factorise it out and divide.

📋 How to collect x onto one side

1
Expand any brackets first, so every x-term is visible.
2
Add or subtract to move every x-term onto one side, and every non-x term onto the other.
3
Factorise: write the x-side as $x (\dots)$ .
4
Divide both sides by whatever is in the bracket.

✏️ Worked example

Make x the subject: $a - b x = d x - a$

1
x is on both sides (as $- b x$ and $d x$ ) — bring them together. Add $b x$ to both sides: $a = d x + b x - a$
2
Add a to both sides, to collect the non-x terms too: $2 a = d x + b x$
3
Factorise x out of the right side: $2 a = x (d + b)$
4
Divide both sides by $(b + d)$ : $x = \frac{2 a}{b + d}$

✅ Answer: $x = \frac{2 a}{b + d}$

⚠️ The classic trap

Moving a term across the = sign without changing its sign — $a - b x = d x - a$ becoming $a = d x + b x + a$ (the second a kept its sign, but it crossed the equals sign, so it must flip). Track every move as "the same operation on both sides," not as "swap the side and keep the sign" — the second one is how this mistake happens.

💡 The idea

A fraction with x underneath — clear it before anything else

You cannot collect terms, expand a bracket, or do anything else around a denominator you haven't gotten rid of yet. So the very first move — before any of the tier-2 collecting — is to cross-multiply: multiply both sides by whatever is underneath the fraction line. That's the exact same cross-multiplying you already use to solve a fraction equation with numbers; the only difference is that the "number" underneath the line now has an x in it. Once the fraction is gone, it's an ordinary x-on-both-sides problem you already know how to finish.

📋 How to clear a fraction with x underneath

1
Multiply BOTH sides by the whole denominator — this is cross-multiplying.
2
Expand any brackets this creates.
3
From here, it's tier 2: collect every x-term on one side, factorise, divide.

✏️ Worked example

Make x the subject: $y = \frac{x + 2}{x - 2}$

1
Multiply both sides by $(x - 2)$ — this clears the fraction completely: $y (x - 2) = x + 2$
2
Expand the bracket: $x y - 2 y = x + 2$
3
Now it's tier 2 — collect x on the left: $x y - x = 2 + 2 y$
4
Factorise and divide: $x (y - 1) = 2 (y + 1) \Rightarrow x = \frac{2 ( y + 1 )}{y - 1}$

✅ Answer: $x = \frac{2 ( y + 1 )}{y - 1}$

⚠️ The classic trap

Cross-multiplying only PART of the top — $y = \frac{x + 2}{x - 2}$ becoming $y x - 2 = x + 2$ instead of $y (x - 2) = x + 2$ . The whole denominator multiplies the whole of the other side — put it in brackets first, expand it second, never skip the bracket.

🤔 Which of these needs cross-multiplying BEFORE anything else?

🎮 Practice · x appears once (plug in the numbers)

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F = \frac{x}{2} - 3

. If

F = 9

, find x.

p = 5 (x - 3)

. If

p = 35

, find x.

y^{2} = x^{2} + 25

. If

y = 13

, find the positive value of x.

P = \frac{x ^{2}}{4} + 7

. If

P = 16

, find the positive value of x.

y - 5 = 2 + \frac{x}{6}

. If

y = 19

, find x.

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🎮 Practice · x on both sides

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4 x - 8 = 3 x

. Find x.

5 x = 20 - 3 x

. Find x.

2 (x + 3) = 4 x

. Find x.

10 - 2 x = 3 x - 10

. Find x.

3 (x - 4) = 2 (6 - x)

. Find x.

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🎮 Practice · fractions with x underneath

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y = \frac{x + 1}{x - 1}

. If

y = 3

, find x.

2 = \frac{x - 2}{3 x}

. Find x.

b = \frac{6 x}{6 + x}

. If

b = 2

, find x.

a = \frac{1 + 2 x}{x - 1}

. If

a = 5

, find x.

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🏅

You finished this lesson!

Three different disguises, one method underneath: undo it, collect it, or clear it — then factorise and divide. The next time a formula looks nothing like

y = m x + c

, it's still just x, waiting to be isolated.

Keep reading — this lesson marks itself done automatically.