GCSE Maths: How to Solve Simultaneous Equations
Introduction
With GCSE Maths Simultaneous Equations, when you have two equations that have two unknowns such as x and y they are referred to as simultaneous equations.
The reason they are called simultaneous equations is because you need to solve them both and at the same time.
The key to dealing with such questions is to make an unknown value to be the same or if an unknown is a unit (equal to 1) and providing that one is positive and the other is negative then you can either add them together or if they are both positive you can then subtract.
But you should always remember that your ultimate aim is to eliminate one of the unknowns and this can be done by either the method of elimination or by using a technique known as substitution.
In order to use the method of substitution it is best if one of the equations has a unit value for an unknown.
GCSE Maths Simultaneous Equations: Example
Take a look at the following question:
Solve the simultaneous equations:
\begin{aligned} & 3 x+2 y=4 \\ & 4 x+5 y=17 \end{aligned}Here you can see that all terms are positive. It is here that you need to decide which variable or unknown you wish to eliminate. Suppose that you decide to opt for eliminating variable y then you need to make these values both the same.
In order to do this the following calculation can be done to each equation i.e. we are going to multiply the first equation by 5 and the second equation by 2:
\begin{aligned} & 15 x+10 y-20 \\ & 8 x+10 y=34 \end{aligned}Following the multiplications there are now two new equations and you will now see that the values for y are both the same. In order to eliminate them equation 4 can be subtracted from equation 3 to give the following:
7 x=-14 \therefore x=-2Quite often, when doing GCSE Maths Simultaneous Equations for Edexcel or any other exam board, students stop at this point. You need to remember that there are two unknowns and when one has been found, you then need to use the method of substitution to find the other. The value of x can be substituted into any of the above equations to find the value of y.
Using equation 1:
\begin{gathered} 3 x+2 y=4 \\ 3(-2)+2 y=4 \\ -6+2 y=4 \\ 2 y=10 \therefore y=5 \end{gathered}Another GCSE Maths Simultaneous Equations Example
Solve the equations:
\begin{aligned} & 3 x+5 y=19 \\ & 4 x-2 y=-18 \end{aligned}Here you will notice that one of the unknowns has a negative value. Again the equations are labelled and it needs to be decided which one of the unknowns should be removed.
Because one of the y terms is negative; it might be easier to eliminate this unknown.
Multiplying the first equation by 2 and and the second equation by 5 will give:
\begin{aligned} & 6 x+10 y=38 \\ & 20 x-10 y=-90 \end{aligned}Now you will be able to see that the y values are the same but one is positive and the other is negative. So in order to eliminate this unknown, it is simply a case of adding equations and 3 and 4 together to give:
26 x=-52 \therefore x=-2Remember there are two unknowns that need to be found and the second can be found using the technique of substitution in any of the above equations.
Using equation 1:
\begin{gathered} 3 x+5 y=19 \\ 3(-2)+5 y=19 \\ -6+5 y=19 \\ 5 y=25 \therefore y=5 \end{gathered}
Question Practice
Question Practice
Try the following GCSE Maths Simultaneous Equation question on your own before looking at the solution.
Solve the simultaneous equations:
\begin{array}{r} 4 x-3 y=11 \\ 10 x+2 y=-1 \end{array}Question Practice Solution
So how did you get on? Hopefully you found that the answer to be x=0.5, y=-3
\begin{array}{r} 4 x-3 y=11 \\ 10 x+2 y=-1 \end{array}First you need to determine which of the unknowns to eliminate. It can be seen that one of the y terms is positive and the other is negative so y will be eliminated by multiplying the first equation by 2 and the second equation by 3 to give:
\begin{aligned} & 8 x-6 y=22 \\ & 30 x+6 y=-3 \end{aligned}These equations can be added together and this will give: 38 x-19 \therefore x-0.5
Finding the y value using equation the first equation:
\begin{gathered} 4 x-3 y-11 \\ 4(0.5)-3 y=11 \\ 2-3 y=11 \\ -3 y=9 \therefore y=-3 \end{gathered}Whether you are doing the foundation or higher gcse maths paper you are going to see a question regarding GCSE Maths simultaneous equations. Always look to see what the signs are. If any are opposite then can you add straight away or do you need to make the numbers the same before you can add?
These are the types of questions that you need to be asking when doing GCSE Maths simultaneous equations. Keep trying the questions within this article to see the various techniques used. You might be able to do the same questions using a slightly different technique such as eliminating a variable which has not been eliminated here, and this actually involves less work. Try and see!