GCSE Maths: How to Solve Equations
Introduction
When it comes to how to solve equations you need to remember a set of patterns.
If you wanted to solve x+3=10
hen what is the value of x? You will be able to spot the answer quite clearly just by looking i.e. 7. But how else can this number also be obtained from the 10 and 3? Here the answer is by subtraction. In other words you subtract the 3 from both sides of the equation.
If you wanted to solve x-3=10
then what is the value of x? You will be able to spot the answer quite clearly just by looking i.e. 13. But how else can this number also be obtained from the 10 and 3? Here the answer is by addition. In other words you add the 3 to both sides of the equation.
If you wanted to solve \frac{x}{2}=3
then what is the value of x? Remember a fraction is another way to write down a division. So what divided by 2 will give 3? The answer is 6. How else can this number 6 be obtained from using the 2 and 3? Here the answer is by multiplication. In other words if you need to solve an algebraic equation that is written in the form of a fraction, always perform a multiplication as discussed here.
If you want to solve 2 x=6
then what is the value of x? Here you need to ask what multiplied by 2 will give the answer of 3? You should be able to spot the answer as being 3, but how else can you obtain this number? Here you can divide 6 by 2. So if you see an equation written as the one shown then you need to perform a division.
How To Solve Equations - An Example
Take a look at the following question:
Here there is an equation which contains a fraction so this is your clue that you need to do multiplication of 6 and 3 to give 18.
Example
Take a look at the following question:
In this type of equation you need to perform a division. The answer will be 547. This is where you need to be careful. 54 and 7 do not simplify. You can perform the calculation on a calculator and you will get a long decimal number. The best thing to do is to leave the answer as \frac{54}{7}. Remember not to let the answer put you off into thinking that something has been done wrong. Simply follow the processes.
More Examples of How To Solve Equations
Example
Take a look at the following question:
This type of question will require you to perform more than one step. Do what you think is the easiest thing first. Looking at the left hand side there is a “2t” and a “-5”. From what was mentioned earlier, you saw that if there was a “negative number” this could be added to the other side.
So add 5 to both sides to give 2t = 14. Now you have an equation that involves a division so the value of t is found by dividing the 14 by 2 to give 7.
Remember you can check your answer.
2 x 7 – 5 = 14 – 5 = 9.
Example
Take a look at the following question:
Here you will notice on the left hand side you have brackets. First you must expand the brackets which will then give:
5 t-15=25Remember, follow the pattern and do what is easiest. Here it would be to add 15 to both sides to give: 5 t=40
Finally you can divide 40 by 5 to give t=8.
Question Practice
Example
Take a look at the following question:
Here you have an x term on both sides of the equal sign and as well as two positive numbers. Here you need to “collect like terms” so have all x terms on one side and all numbers on the other.
It does not matter where the xterm is, but it will be helpful if you ensure that it always stays positive. Because 13 x is bigger than 11 x it would make sense to subtract 11 x from both sides. This would then give: 2 x+1=9
Now subtract the 1 from both sides to give: 2 x=8
Finally divide the 8 by 2 to give x=4
When it comes to how to solve equations, many students do struggle, but if you follow the techniques and suggestions in this article then there should be no question that phases you!
In future articles we will look at more complex equations, but you need to remember that no matter how complex the question, the principle technique of solving the equations is the same as what is detailed here.