Algebra 5

A linear equation is an equation involving one unknown, often denoted by x, and one or more numbers (the linear part of the expression means that there are no power terms ie no x² , x³ etc). Some examples are shown below:

3x - 1 = 4

7 = 2 - x

(x/100) + 10 = 20.

There may be more than one x term. For example, in the equation below x appeae s twice, once on the left hand side and once on the right hand side.

x - 3 = 3x + 7

And there may be brackets. For example, in the equation below there are two terms in brackets.

x - 2(x - 1) = 1 - 4(x + 1)

We need to be able to solve linear equations ie find the value of x that make them true. So, for example, given the equation

3x - 1 = 4

we need to find the value of x that makes the equation true. In general, there will only be one such value ie only one solution.

The following example illustrates the general method of solution.

Example

Solve the equation

x + 3(x + 1) = 2x

Step 1: multiply out any brackets

x + 3x + 3 = 2x

Step 2: collect like terms. Here we could put the x values on the left hand side of the equation and the other values on the right hand side. So the equation becomes

x + 3x - 2x = -3 (note that the 2x becomes -2x when it is moved and the 3 becomes -3. When you move something from one side of the equation to the other side you must change the sign.)

Step 3: add like terms

2x = -3

Step 4: divide both sides of the equation by 2

x = -3/2

Step 5: check (not compulsory, but a good idea if you have the time or are a bit unsure). If we plug or substitute x = -3/2 into the original equation it should work.

Example 1 (Foundation Nov 15 Paper 1 Q5(d) (i) )

Solve 3y = 24 (Answer y = 8)

Example 2 (Foundation Nov 15 Paper 1 Q5(d) (ii) )

Solve x - 4 = 11 (Answer x = 15)

Example 3 (Foundation Nov 14 Paper 1 Q9(b) )

Solve 5t - 6 = 9 (Answer t = 3)

Example 4 (Foundation May 15 Paper 1 Q5(d) (iii) )

Solve 9 - t = 7 (Answer t = 2)

Example 5 (Foundation Nov 15 Paper 1 Q12)

Solve 8x - 9 = 21 + 5x (Answer x = 10)

Example 6 (Foundation May 15 Paper 2 Q14)

Solve the following equation

2(30 - x) = 44 (Answer x = 8)