Solving Simple Equations

The aim of this document is to provide a short, self assessment programme for students who wish to acquire a basic competence at solving simple equations.

In this section we shall look at some simple equations and the methods used to find their solution. There are four basic rules:

Rule 1 An equal quantity may be added to both sides of an equation.

Rule 2 An equal quantity may be subtracted from both sides of an equation.

Rule 3 An equal quantity may multiply both sides of an equation.

Rule 4 An equal, non-zero quantity may divide both sides of an equation.

The application of these rules is illustrated in the following examples.

Example

Solve the equations

Solution

(a) By Rule 1 we may add 8 to both sides:

3 x - 8 + 8 = x + 10 + 8 i .e. x = x + 18.

By Rule 2 we may subtract x from both sides:

3 x - x = x + 18 - x i .e. 2 x = 18.

Finally, by Rule 4 we may divide both sides by 2 giving x = 9 .

(b) By Rule 3 we may multiply both sides by 2,

It is always good to check that the solution is correct by substituting the value into both sides of the equation. In Example 1 (a), by substituting x = 9 into the left hand side of the equation we see that x - 8 = 3 × 9 - 8 = 19. Substituting x = 9 into the right hand side of the equation gives x + 10 = 9 + 10 = 19. Since both sides of the equation are equal when x = 9, it is a correct solution. In this case it is the only solution to the equation but it is important to note that some equations have more than one solution.

Exercise

(a) 3 x = 18 , (b) 7 x = -14 (c) -2 x = -10 (d) 28 x = 35 (e) 5 x - 3 x - 12 x = 29 - 2 - 7 (f)

Solution

(a) Dividing both sides by 3 gives

or x = 6 .

(b) Dividing both sides by 7 gives

or x = - 2 .

(c) Dividing both sides by - 2 gives

or x = 5 .

(d) Here 7 is the highest common factor of 28 and 35. First let us divide both sides by this.

28 x = 35

4x = 5

Now divide both sides by 4.

The solution is thus x = 5 / 4 .

(e) First let us simplify both sides. The left hand side is

5 x - 3 x - 12 x = 5 x - 15 x = - 10 x .

The right hand side is

29 - 2 - 7 = 29 - 9 = 20 .

The original equation is thus - 10 x = 20

and the solution to this is obtained by dividing both sides of the equation by - 10.

so that x = - 2 .

(f) In this case we must multiply both sides by 5.

and the solution in this case is x = - 15 .

Try the following short quizzes.

Quiz 1

Which of the following is the solution to the equation

8 x + 5 x - 3 x = 17 - 9 + 22 ?

(a) 2 (b) -2 (c) 3 (d) -3

Solution:

Simplify both sides first:

8 x + 5 x - 3 x = 13 x - 3 x = 10 x .

17 - 9 + 22 = 8 + 22 = 30 .

The equation to be solved is thus 10x = 30 and this clearly has solution x = 3 .

Quiz 2

Which of the following is the solution to the equation

x - 13 x = 3 x - 6 ?

(a) 2 5 (b) - 1 5 (c) 1 3 (d) - 6 17

Solution:

x - 13 x = 3 x - 6

- 12 x = 3 x - 6 .

0 = 12 x + 3 x - 6

15 x - 6 = 0

15 x = 6