Substitution and rearranging formulae

To substitute, replace each letter with its value, then work it out using BIDMAS.

1. Write the expression with brackets where each letter was.
2. Put the numbers in the brackets.
3. Work it out, taking care with negative numbers.

Example: find 3a² − 2b when a = −4 and b = 5.

1. 3 × (−4)² − 2 × (5)
2. = 3 × 16 − 10
3. = 38

Example: v = u + at. Find v when u = 12, a = −3 and t = 2.5.

1. v = 12 + (−3) × 2.5
2. = 12 − 7.5
3. = 4.5

The subject of a formula is the letter on its own on one side, such as v in v = u + at. To change the subject, undo what has been done to it, doing the same to both sides each time.

1. Undo additions and subtractions.
2. Undo multiplications and divisions.
3. Undo powers and roots.

Example: make x the subject of y = 5x − 3.

1. Add 3 to both sides: y + 3 = 5x
2. Divide both sides by 5: x = (y + 3)/5

Harder example: make r the subject of A = πr².

1. Divide both sides by π: r² = A/π
2. Square root both sides: r = √(A/π)

The area of a circle, A = πr², is on the formulae sheet; you still need to rearrange it.

If the new subject appears in more than one term, gather those terms together and factorise.

1. Clear any fractions by multiplying.
2. Expand brackets.
3. Get every term with the subject on one side, and the rest on the other.
4. Factorise the subject out.
5. Divide by the bracket.

Example: make x the subject of ax + 3 = bx + 7.

1. ax − bx = 7 − 3
2. x(a − b) = 4
3. x = 4/(a − b)

Harder example: make x the subject of y = (x + 2)/(x − 3).

1. Multiply by (x − 3): y(x − 3) = x + 2
2. Expand: xy − 3y = x + 2
3. Rearrange: xy − x = 3y + 2
4. Factorise: x(y − 1) = 3y + 2
5. x = (3y + 2)/(y − 1)