Equations can often be solved by guessing.

"I think of a number and add 6 to it."
"The result is 10. "
"What is the number?"

Most people can easily work out that the number is 4.
In mathematics, we would write an equation x + 6 = 10, where x stands for the number. We would then solve the equation to find the value of the number (x).
This sounds a long way of doing things but many equations are more difficult to solve than the one above.
There are several types of equations and several methods of solving them.
When solving equations each step should be written on a new line, and the equals signs should be kept directly underneath each other.
Linear Equations
One Step Equations 
Examples 
Answers 
x + 7 = 15

x + 7 = 15
x + 7 − 7 = 15 − 7 (subtract 7 from both sides)
x = 8

g − 3 = 12

g − 3 = 12
g − 3 + 3 = 12 + 3 (add 3 to both sides)
g = 15

3w = 15

3w = 15
^{3w}⁄_{3} = ^{15}⁄_{3} (divide both sides by 3)
w = 5

^{k}⁄_{4} = 5

^{k}⁄_{4} = 5
^{k}⁄_{4} × 4 = 5 × 4 (multiply both sides by 4)
k = 20

Two Step Equations

Examples

Answers


3p + 10 = 22

3p + 10 = 22
3p + 10 − 10 = 22 − 10
3p = 12
p = 4

(subtract 10 from both sides)
(divide both sides by 3)

2g − 7 = 13

2g − 7 = 13
2g − 7 + 7 = 13 + 7
2g = 20
g = 10

(add 7 to both sides)
(divide both sides by 2)

The answer should then be checked by substituting it back into the equation to make sure that both sides have the same value. This step can be done mentally. e.g. For 3w = 15. The solution is w = 5. Check 3 × 5 = 15.
Equations with Brackets
Generally, it is better to expand any brackets first.
e.g. 
3(x + 7) = 24



3x + 21 = 24

(expand brackets) 

3x + 21 − 21 = 24 − 21

(subtract 21 from both sides) 

3x = 3

(divide both sides by 3) 

x = 1


Problem Solving
Many problems can be solved by making equations and solving them.
The general way to approach these problems is:
 Assign a variable or letter to the unknown quantity in the problem.
 Make up an equation from the information in the problem.
 Solve the equation.
Example 
Answer 
Hemi thinks of a number, doubles it and subtracts 8.
The result is 32.
What is his number?

Let the unknown number be x.
The equation is:
2x − 8 = 32
Solving the equation:
2x − 8 = 32
2x = 32 + 8
2x = 40
x = 20
Hemi's number is 20

