Using and Rearranging a Formula (Subject Appears Once)

You are at a taxi rank with eleven pounds in your pocket. Picture a fare that works like this: three pounds to start, then one pound fifty for every mile you travel. Written as a formula, that is C equals three plus one point five m, where C is the cost in pounds and m is the number of miles. That formula is built to answer one question. You feed it a distance, and it hands you back a cost. Your problem points the other way. You already know what you can afford, and the thing you want out is the distance. Turning a formula round so it answers the question you actually have is the skill in this video, and it is one of the most mechanical things in algebra.

Start with the half that looks too easy: using a formula somebody has already handed you.

When a formula is given to you, on a formula sheet or printed inside the question, your first job is to copy it exactly as written.

Here is one. The perimeter of a rectangle is P equals two l plus two w, with l the length and w the width.

Copy it down, then read it back against the original, character by character. Two l, plus two w. That is what it says, so that is what you use.

Now put the numbers in. The length is seven centimetres and the width is four centimetres.

Three ways of writing that substitution are on screen. A: two times seven, plus two times four. B: twenty-seven plus twenty-four. C: two plus seven plus two plus four. Only one of them is what the formula actually says.

Take your pick. I'll wait.

The answer is A. Two l is a multiplication, two lots of l, not the digit two written next to the letter and not two added on. So P is fourteen plus eight, which is twenty-two centimetres.

Every formula in this video is given to you. Recalling one from memory with nothing printed on the page is the video called Recalling Circle, Pythagoras and Trig Formulae.

That whole half runs on one habit: copy it exactly, then put the numbers where the letters were.

The bigger skill is rearranging: changing which letter the formula hands you at the end.

Making t the subject just means getting t alone on one side, with everything else on the other.

Take k equals five t minus two. Look at what that formula does to t, in order. It multiplies t by five. Then it subtracts two from the result.

That order is what tells you where to start, because you are going to undo it backwards.

Last on, first off. It is getting undressed at the end of the day: the coat went on last, so the coat comes off first.

The subtract two went on last, so it comes off first. Add two to both sides, and write that line down.

That leaves k plus two equals five t. The minus two has gone, and nothing else on that side has changed.

One thing is still attached to t, and that is the multiply by five. Undo it by dividing both sides by five.

And there it is. t equals k plus two, all over five.

The k plus two stays bracketed together, because the whole of that side got divided by five, not just the k.

Both of those moves were done to both sides. That is the same balancing used to solve an equation for a number, taught in the video called Solving Linear Equations by Balancing.

The one difference is that your answer comes out as letters rather than a number. A rearranged formula is still a formula.

The same move handles a formula with only one step in it, including one borrowed from another subject.

A formula sheet gives force as F equals m a, so force is mass times acceleration. Your job is to make a the subject.

Two candidates are on screen. Is a equal to F over m, or is it m over F?

Have a think. I'll wait.

The answer is a equals F over m. The a is multiplied by m, so you divide both sides by m, and the m cancels away on the right.

The other version is what you get by shuffling letters around instead of asking what is attached to the one you want.

One step or two, you ask the same thing every time: what has been done to this letter, and in what order.

Your turn now, on a formula built the same way as the one just worked through.

Make x the subject of y equals four x minus seven. Write the add seven step and the divide by four step on two separate lines.

Pause it there and work it out. I'll wait.

Seven was taken away last, so it comes off first. Add seven to both sides, and you have y plus seven equals four x. Then x is multiplied by four, so divide both sides by four. x equals y plus seven, all over four.

If both of those lines are on your page, you have done exactly what this topic is asking for.

Now the exam side of it, starting with something that goes wrong before any maths does.

Here is one line from an examiner report, about students working with formulae they had been given.

Sometimes students forgot that the formula sheet on page 2 existed or copied the formulae incorrectly, which was a shame.

Forgetting the sheet throws away a formula you never had to remember. Copying it wrong quietly breaks everything built on top of it. So the check takes seconds. Find the sheet, copy the formula, read it back against the print before you touch a number.

The second line is about the rearranging itself, from a report on a question where a formula had to be rearranged.

This was poorly answered. Few candidates showed complete stages in their rearrangement and many went straight to a solution, often w equals two P plus two h. Some candidates wrote divide by two, but instead subtracted two from individual terms.

Read that last sentence again. Dividing by two and subtracting two are different moves, and going straight to an answer leaves nowhere to catch the swap. That is why each stage gets its own line. A written line is a checkpoint, and working done in your head has no checkpoints.

This method has an edge to it. If the letter you want appears twice, or sits under a power or a square root, it needs an extra idea on top. And if you have to build the formula yourself from a written description, that is the video called Writing an Expression or Formula from a Context.

Time to gather this into something you can carry into the exam hall. Given a formula, copy it exactly, then put the numbers where the letters were. Two l means two times l. Rearranging: last on, first off. Find what was done to your letter last, and undo that first. In a formula shaped like five t minus two, multiply then subtract, the subtract went on last, so it comes off first. Do every move to both sides, and give every stage its own line on the page.

Next in the chapter: Rearranging Harder Formulae, for when the letter you want appears twice, or sits under a power or a root.

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