# 2.11 Manipulation of equations

An equation is a mathematical expression that shows the relationship between numbers through the use of the equal sign. An example of a simple equation might be 3 + 2 = 5.

An equation could also be in the form of £5,000 - £2,000 = £3,000 to express mathematically the accounting fact that sales of £5,000 minus costs of £2,000 equals a profit of £3,000. (An important aspect of mathematics, but not of accounting, is that £5,000 - £2,000 = £3,000 can be simplified as 5 – 2 = 3.) Another well-known use in accounting of an equation is the accounting equation, which you will learn in more detail next week. The accounting equation is the basis of the balance sheet, which you will learn how to produce in Week 4.

## Box 5 The relationship between numbers in the accounting equation

The accounting equation states that Assets (A) = Capital (C) + Liabilities (L). Such an equation, which can also be abbreviated as A = C + L, can be stated in financial terms for a particular business at a particular time:

£100,000 of Assets (A) = £80,000 of Capital (C) + £20,000 of Liabilities (L)

The accounting equation can be expressed in the different forms below, which are all correct for the example of our business with assets of £100,000:

A = C + L or £100,000 = £80,000 + £20,000 (accounting equation)

C + L = A or £80,000 + £20,000 = £100,000

C = A - L or £80,000 = £100,000 - £20,000

L = A - C or £20,000 = £100,000 - £80,000

A – L = C or £100,000 - £20,000 = £80,000

A – C = L or £100,000 - £80,000 = £20,000

The simple equation of 3 + 2 = 5 is true as long as each of the three numbers does not change. If one number is hidden, as long as the other two numbers are known in our example, then the hidden third number can be worked out easily. For example, if ‘3’ is hidden in 3 + 2 = 5, we know that this number must be 3 in order to make the equation true.

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 in step 1 above i.e.:

*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* = 3

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 solve the correct number for the algebraic letter.

Operation | Inverse | Equation | Manipulation to solve 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. (By convention, the number is always put before the letter i.e. 3a not a3).

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