Formulas: Substitute and Rearrange

To substitute into a formula, replace each letter with its value, and put every value in brackets, especially a negative one. In v equals u plus a t, with u equal to twenty, a equal to negative three and t equal to four, write v equals twenty plus, in brackets, negative three, times four. a t means a times t, and multiplication comes before addition, so work out negative three times four first: negative twelve. Then twenty plus negative twelve is eight. So v equals eight.

Here is the area of a trapezium: A equals one half, times a plus b in brackets, times h. With a equal to six, b equal to ten and h equal to five, work out the bracket first: six plus ten is sixteen. Half of sixteen is eight. Eight times the height, five, is forty. So the area is forty square units. The bracket is the whole point: the two parallel sides are added before anything else happens to them.

To change the subject, look at how the formula builds the answer from the letter you want. Starting from a, the formula multiplies by t and then adds u to get v. To get back to a, undo those steps in reverse order: first subtract u from both sides, giving v minus u equals a t, then divide both sides by t. So a equals v minus u, all over t. Check with numbers: if u is two, a is three and t is four, then v is fourteen, and fourteen minus two, over four, is three.

Some subjects take several steps. To make b the subject of A equals a half, a plus b, times h, first clear the half by multiplying both sides by two: two A equals a plus b, times h. The bracket a plus b stays together as one group while you divide both sides by h: two A over h equals a plus b. Only now can you subtract a: b equals two A over h, minus a. Check with the trapezium from before: two times forty over five is sixteen, minus six is ten, and b was ten.

Watch the sign when the subject has a minus in front of it. In y equals five minus three x, subtract five from both sides: y minus five equals negative three x. Dividing by negative three gives x equals y minus five over negative three, which tidies to five minus y, all over three. A neater way is to add three x to both sides first, so the x-term is positive: three x plus y equals five, then three x equals five minus y. Either way, x equals five minus y, over three. Check: x equals one gives y equals two, and five minus two, over three, is one. Quiz and exercises now.