1.10 Manipulation of equations and formulae
The final activity in developing your numerical skills is to revise the manipulation of simple equations.
Being able to understand and express the Accounting Equation in different forms is crucial to understanding a fundamental accounting concept (the dual aspect concept) and the principal financial statements (the profit and loss account and the balance sheet). You will learn more about the Accounting Equation in sections 2 and 3.
An equation is a mathematical expression which shows the relationship between numbers through the use of the equal sign. An example of a simple equation might be 2 + 3 = 5.
A special type of equation is an algebraic equation where a letter, say ‘x’, represents a number, i.e. in x + 2 = 5, ‘x’ represents 3 in order to make the equation true.
Algebraic equations are solved by manipulating the equation so that the letter stands on its own. This is achieved in the equation x + 2 = 5 by the following two steps.
- x = 5 – 2
- x = 3
The principal rule of manipulating equations is whatever is done to one side of the equal side must also be done to the other, as was shown above.
x = 5 – 2 is achieved by subtracting 2 from both sides of the equation x + 2 = 5, i.e.:
- x + 2 – 2 = 5 – 2
- x = 5 – 2
Manipulating an equation to get the algebraic letter to stand on its own involves ‘undoing’ the equation by using the inverse or opposite of the original operation. In the example of x + 2 = 5, the operation of adding 2 must be undone by subtracting 2 from either side of the equal sign.
The following table shows a number of examples of how equations are manipulated to obtain the correct number for the algebraic letter.
|add 7||subtract 7||a + 7 = 9|
|a + 7 – 7 = 9 – 7|
|a = 2|
|subtract 5||add 5||b – 5 = 6|
|b – 5 + 5 = 6 + 5|
|b = 11|
|multiply by 3||divide by 3 (or multiply by 1/3)||c x 3 = 18|
|c x 3 / 3 = 18 / 3|
|c = 6|
|divide by 6||multiply by 6||d / 6 = 2|
|d / 6 x 6 = 2 x 6|
|d = 12|
An equation such as a x 3 = 12 can also be expressed as a3 = 12 or 3a = 12, i.e., if an algebraic letter is placed directly next to a number in an equation it means that the letter is to be multiplied by the number.
The correct number for the algebraic letter ‘a’ in the equation 3a = 12 will be obtained thus:
- 3a = 12
- 3a / 3 = 12 / 3
- a = 4
Manipulating or rearranging formulae involves the same process as manipulating or rearranging equations.
A formula is simply an equation that states a fact or rule such as S = D / T or Speed is equal to Distance divided by Time.
In the formula S = D / T, S is the subject of the formula. (This simply means that S stands on its own and is determined by the other parts of the formula. By convention the subject is always placed on the left-hand side of the equal sign, although S = D / T means the same as D / T = S)
As we learnt to rearrange or manipulate an equation, the formula S = D / T can also be manipulated to make D or T the subject.
S = D / T
D / T = S (turning the formula around)
D = S x T (multiplying both sides of the formula by T)
Or, from D = S x T
D / S = T (dividing both sides of the formula by S)
T = D / S (turning the formula around)
Solve the following algebraic equations below.
(i) c + 9 = 11
(i) c =2
(ii) a – 15 = 21
(ii) a = 36
(iii) d x 7 = 63
(iii) d =9
(iv) b / 13 = 13
(iv) b = 169
Rearrange the formula h = 3dy – r to make:
(i) r the subject
(i) h = 3dy – r
h + r = 3dy
r = 3dy – h
(ii) y the subject
(ii) h = 3dy – r
h + r = 3dy
3dy = h + r
y = (h + r) / 3d (Hint: did you remember to use brackets?)