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2.6 Deriving and using formulae

2.6 Deriving and using formulae

A formula is a mathematical rule that shows the relationship between two or more quantities (variables).
It is a rule that can be written in letters or words. The plural of formula is formulae.
You can write, or derive, your own formulae to help you solve problems.
When you substitute numbers into formulae and expressions,
remember the order of operations, BIDMAS. Brackets and Indices Examples of indices are
3
2
3
2
must be worked out before Divisions and Multiplications. Additions 2 , 5 , 4 and 7 .
and Subtractions are always worked out last.
Worked example 2.6

a Work out the value of the expression 2x + 4y when x = 5 and y = −2.
b Work out the value of the expression 3x + 4 when x = 10.
2
c Write a formula for the number of hours in any number of days, using: i words ii letters.
d Use the formula in part c to work out the number of hours in 7 days.

a 2 × 5 + 4 × −2 Substitute x = 5 and y = −2 into the expression.
= 10 + −8 Work out 2 × 5 and 4 × −2.
= 10 − 8 = 2 Adding −8 is the same as subtracting 8.
2
b 3 × 10 + 4 Substitute x = 10 into the expression.
2
= 3 × 100 + 4 Work out 10 first.
= 300 + 4 = 304 Work out the multiplication before the addition.
c i hours = 24 × days There are 24 hours in every day.
ii h = 24d Use h for hours and d for days.
d h = 24 × 7 = 168 Substitute d = 7 into the formula and work out the value of h.

✦ Exercise 2.6

1 Work out the value of each expression for the given values.
a p + 5 when p = −3 b q − 6 when q = 4 c 6h when h = −3
j
d when j = −20 e a + b when a = 6 and b = −3 f c – d when c = 25 and d = 32
4
g 8m − 5 when m = −2 h 3z + v when z = 8 and v = −20 i 2x + 3y when x = 4 and y = 5
u p q
j 20 − 3n when n = 9 k 5 when u = 4 l + when p = 30 and q = −8
2 5 2

2 Work out the value of each expression.
2
2
2
2
a x + 5 when x = 4 b 10 − y when y = 5 c g + h when g = 3 and h = 6
2
2
2
2
d m − n when m = 7 and n = 8 e 4k when k = 2 f 2d + 1 when d = 5
3
3
3
g r when r = 1 h 2y when y = 3 i x − 5 when x = 2 Remember that r
3
3
j 20 − w when w = 4 k y 2 when y = 4 l p 3 when p = 10 means r × r × r.
2 5
2 Sequences, expressions and formulae 27