## Formulae

A formula is a general rule or relation, written as an algebraic equation. There are all types of formulae for calculating various quantities in many subject areas.

### Substitution

Substitution involves putting given values into a formula in place of the variables (written as letters).

e.g. is the formula for the area of a trapezium.

If a = 5 cm, b = 7 cm and h = 10 cm A = 60 cm2

### Changing the Subject

The subject of a formula is the variable or letter that stands alone.

 e.g. A is the subject x + y + z = V V is the subject

To make a different variable the subject of a formula, the usual algebraic techniques can be used.

Example 1

To make b the subject of  Example 2

To make x the subject of A = x2

Take the square root of both sides

x = ±√A

When you have solved an equation, you can always check your answers by substituting the values back into the equation.

(a) Solve x2 − 4x − 5 = 0

Factorise to (x − 5)(x + 1) = 0

Either: x − 5 = 0 or x + 1 = 0

x = 5 or x = -1

To check:

 Substitute x = 5 into the equation. x2 − 4x − 5 = 0 52 − 4.5 − 5 = 0 25 − 20 − 5 = 0 0 = 0 (This shows the solution x = 5 is correct.) Substitute x = -1 into the equation. x2 − 4x − 5 = 0 (-1)2 − 4.(-1) − 5 = 0 1 + 4 − 5 = 0 0 = 0 (This shows the solution x = -1 is correct.)