GCSE · Maths · AQA · Spec 8300

Standard mathematical formulae and rearranging

Every formula is a little machine: a number goes in, gets worked on in a set order, and an answer comes out. Today you'll run it forwards and backwards.

Build a formula, then run it backwards

1Start with m23456

Stage 1 of 6: Start with m. paused

Start with m · 1/6scrub forwards, then back →

Take a 6-mile journey, so m = 6. m is the letter we'll want on its own later, so keep an eye on what happens to it.

A taxi charges £3 to get in, then £2 for every mile. For a journey of m miles, the cost is £C.

A formula from science, given in words

kinetic energy = ½ × mass × speed² → E = ½mv²

E is the kinetic energy (in joules), m is the mass (in kg) and v is the speed (in m/s). The formula gives you E. Letters written side by side are multiplied, so ½mv² means ½ × m × v², and the little ² belongs to v alone: only the speed is squared.

The words and the symbols say exactly the same thing. To translate, swap each quantity for its letter. You'll meet formulae from other subjects this way, and the maths you use on them is the same.

Substituting

You choose the next line

The formula y = 2x² − 3x is given. Find the value of y when x = −4.

  1. y = 2x² − 3x, with x = −4
  2. missing step
Which line is step 2?

Changing the subject

Which step comes first?

Put these steps in the order that makes x the subject of y = (4x − 1)/5 + 2. They start in the order you meet them reading left to right.

1 · Do first4 · Do last

  1. Divide both sides by 4

  2. Add 1 to both sides

  3. Multiply both sides by 5

  4. Subtract 2 from both sides

Watch out: Starting with ÷ 4 isn't against the rules, but it doesn't free x. You'd have to divide the fraction and the + 2 by 4, and x would still be stuck inside the fraction.

Rearrange, then substitute

Where does this answer go wrong?

The formula y = 4x − 8 is given. Make x the subject, then find x when y = 20.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Feed numbers in and you're substituting. Undo it step by step and you've changed the subject.

What you need to know

  • A formula shows how quantities are related. The subject is the letter on its own on one side, like C in C = 2m + 3.
  • A formula given in words can be written in symbols by swapping each quantity for its letter. Letters written side by side are multiplied.
  • To substitute, replace every letter with its value, putting negatives in brackets. Then work out brackets first, then powers, then × and ÷, then + and −.
  • To change the subject, list what was done to the letter you want, in order, then undo those operations in reverse: last on, first off.
  • Each undo is done to the whole of both sides. If you are about to divide a side with more than one term, bracket it first.

The big picture

A formula is a relationship between quantities, and it can be written in words or in symbols. To substitute, replace each letter with its value (negatives in brackets) and work it out in the right order: brackets, then powers, then multiplying and dividing, then adding and subtracting. To change the subject, find what was done to the letter you want and in what order, then undo those operations in reverse, doing each one to the whole of both sides.

Key points

1Forwards and backwards: C = 2m + 3 does × 2 then + 3 to m, so m = (C − 3)/2 undoes them as − 3 then ÷ 2.
2In 2x² only the x is squared. With x = −4, 2x² = 2 × (−4)² = 2 × 16 = 32.
3A negative squared is positive: (−4)² = 16, but −4² = −16 because only the 4 is squared.
4When the letter is inside a fraction, undo from the outside in: first clear anything added to or taken away from the fraction, then multiply to remove the denominator.
5Checking a rearrangement: choose a value, run it through the original formula, then put the result into your new formula. You should get your value back.

Worked example

Problem

The formula E = ½mv² is given. Make m the subject, then find m when E = 400 and v = 10.

⚠ Watch out

Dividing only part of a side. From P = 3a + 6, subtracting 6 gives P − 6 = 3a. Writing a = P/3 − 6 divides the P but not the 6. The whole of P − 6 must be divided by 3: a = (P − 6)/3.

🧠

Memory hook

Socks on, then shoes on. To undo it, the shoes come off first. Changing the subject works the same way: last on, first off.

✓

Check yourself

Make t the subject of v = u + 5t. Then find t when v = 32 and u = 12. (Answer: t = (v − u)/5, so t = 20 ÷ 5 = 4.)

Flashcards

(9)
What is the subject of a formula?
The letter on its own on one side. In C = 2m + 3, the subject is C.
What does 'change the subject' mean?
Rearrange the formula so a different letter is on its own. The relationship stays exactly the same.
In what order do you undo the operations when changing the subject?
The reverse of the order they were done to that letter. Last on, first off.
What does 'do the same to both sides' really mean?
Do it to the whole of each side. Before dividing a side with more than one term, put it in a bracket.
Why put a negative number in brackets when you substitute it?
So a power applies to the whole number, sign included: (−3)² = 9, but −3² = −9.
In 3a², what gets squared?
Only the a. 3a² means 3 × a². Squaring the 3 as well would need (3a)².
After substituting, what order do you work in?
Brackets, then powers, then × and ÷, then + and −.
How do you turn a formula in words into symbols?
Swap each quantity for its letter. 'Times' is usually shown by writing the letters side by side, so 'length × width' becomes lw.
How can you check a rearranged formula?
Pick a value, put it into the original formula, then feed the answer into your new formula. You should get your value back.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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