Linear Equations

A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. In simple terms it is a mathematical sentence in which you can see only one letter (which might appear more than once) but there will be no powers (squared, cubed etc). Here is an example of a simple linear equation:

2x + 7 = 15

This equation can be "solved" to find which value is represented by the letter x.

The eQuation Generator above can make up unlimited equations for you to practise solving. You can change the options so that one of five different types of equation is displayed, choose the level of difficulty and decide whether the answers should be positive, negative or a mixture of both.

The difficulty levels change the structure of the equations rather than just making the numbers bigger. Simple equations are set out in the most familiar way with small numbers. Medium equations may be written the other way round, include negative numbers and use a wider variety of forms. Hard equations need more careful thought, with negative coefficients, subtracted brackets and more than one fraction. Within each level the examples start gently and become slightly more challenging each time you press the Next button. The Restart button takes you back to the start of the level but will present different equations.

Click the Answer button to reveal the first line of working, then keep clicking to reveal each further line until the solution is reached. If you would just like to see the answer, click on the lower panel.

Here are examples showing a good way to solve equations by thinking of the two sides of the equation as two sides of a balance. The equation will remain balanced only if you do the same thing (multiply, divide add or subtract) to both sides.

Type 1

\(3x=12\)
Divide both sides by 3
\(x= 4\)

By doing the same thing to both sides of the equation you can find what one x is equal to.

Self marking exercise

Type 2

\(4x - 3 = 13\)
Add 3 to both sides
\(4x=16\)
Divide both sides by 4
\(x = 4\)

Self marking exercise

Type 3

\(5x + 3 = 3x + 15\)
Subtract 3 from both sides
\(5x = 3x+ 12\)
Subtract 3x from both sides
\(2x= 12\)
Divide both sides by 2
\(x = 6\)

Self marking exercise

Type 4

\(2(3x - 4) + 1 = 5\)
Subtract 1 from both sides
\(2(3x - 4) = 4\)
Divide both sides by 2
\(3x - 4 = 2\)
Add 4 to both sides
\(3x = 6\)
Divide both sides by 3
\(x = 2\)

Another method:

\(2(3x - 4) + 1 = 5\)
First multiply out (expand) the brackets
\(6x - 8 + 1 = 5\)
Collect together like terms
\(6x - 7 = 5\)
Add 7 to both sides
\(6x = 12\)
Divide both sides by 6
\(x = 2\)

Self marking exercise

Type 5

\( \frac{2x+3}{5} + 7 = \frac{3x+12}{3} \)
Multiply both sides by 15 (the lowest common multiple of the denominators)
\(6x+9+105=15x+60\)
Collect together like terms
\(6x+114=15x+60\)
Subtract 60 from both sides
\(6x+54=15x\)
Subtract 6x from both sides
\(54=9x\)
Divide both sides by 9
\(x = 6\)

Self marking exercise

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