Matemáticas
What Is the Subject of a Formula?
In v = u + at the subject is v. Making a the subject turns it into a = (v − u)/t — the same relationship, rearranged to answer a different question.
Subject of a formula
The subject of a formula is the single letter that stands alone on one side of the equals sign, with every other letter on the other side.
- También llamado
- Changing the subject, Transposing a formula, Rearranging formulae
- Dónde lo encuentra el alumnado
- Grade 8 mathematics with one-step rearrangements, and at GCSE and IGCSE on both tiers, including versions where the wanted letter appears twice.
La respuesta corta
The subject of a formula is the letter standing alone on one side of the equals sign, and it is the quantity the formula gives you. In v = u + at, v is the subject. 'Make a the subject' means rewriting it as a = (v − u)/t, so that the formula now hands you a instead.
Un ejemplo
v = u + at → a = (v − u)/t
Both lines say the same thing about the same four quantities. The first is arranged to answer 'what is the final speed', the second to answer 'what was the acceleration'. Nothing has been discovered by rearranging; the formula has simply been pointed at a different question.
The subject is what the formula is for
A formula is written with a particular output in mind, and the subject names it. v = u + at is set up for someone who knows the starting speed, acceleration and time and wants the final speed. Everything on the right is an input; the letter on the left is what comes out.
So 'make a the subject' is not a pointless exercise in moving letters around. It converts a formula built for one purpose into one built for another, and it is the difference between a formula you can use directly and one you have to solve every time you need an answer.
One formula, four subjects
Every letter in a formula can take its turn as the subject, and each version is a legitimate formula in its own right. These four lines all state the same relationship between speed, acceleration and time.
Given values for three of the quantities, any of them can be used — but if you need a for twenty different sets of readings, rearranging once and substituting twenty times is far less work than solving twenty equations.
- v = u + at — gives the final speed
- u = v − at — gives the starting speed
- a = (v − u)/t — gives the acceleration
- t = (v − u)/a — gives the time
When the letter you want appears more than once
If the wanted letter shows up once, isolating it is a matter of undoing the operations around it. If it shows up twice, that route is closed: in y = (x + 3)/(x − 1) there is no single x to isolate, because moving one of them leaves the other behind.
These are a genuinely different kind of question. Every term containing x has to be gathered on one side first, and x then taken out as a common factor, before there is anything to divide by. Questions that say 'make x the subject' where x appears twice are testing factorising as much as rearranging, which is why they sit on the Higher tier.
Preguntas frecuentes
What does 'make x the subject' mean?
Rewrite the formula so that x sits alone on one side and everything else is on the other. You have finished when the line reads x = something, and that something contains no x at all. If an x is still lurking on the right-hand side, the rearrangement is not complete.
Does the subject have to be on the left?
By convention it is written on the left, and that is how answers are usually presented, but a = (v − u)/t and (v − u)/t = a say exactly the same thing. What matters is that the letter is alone on one side and appears nowhere on the other.
Why change the subject instead of substituting first?
Substituting first is fine for a single value and often quicker. Rearranging pays off when you need the same quantity repeatedly, when the numbers are awkward, or when the question asks for an answer in terms of the other letters rather than a number — in which case there are no values to substitute.
Is a formula the same as an equation?
A formula is a particular kind of equation: one with a subject, relating quantities that each have a meaning, such as A = lw or v = u + at. Both have equals signs and both can be rearranged, but a formula is meant to be used with values put in, not solved for a mystery number.
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