How do we solve simultaneous equations, say for example x + 4y = 20 and 2x - 2y = 10 ?

When it comes to simultaneous equations, there are two methods that we can use to solve them. The first method is called substitution, where we make one of the variables the subject in one of our equations and "plug it" into the other equation. For our example, we have :

(1) x + 4y = 20


(2) 2x - 2y = 10 

We'll take equation (2) and make x the subject of it. We would first add 2y on either side of the equals sign to get rid of it on the left side:

2x = 10 +2y

We now nearly have an equation where the subject is x; all we need to do is divide it by 2.

x = 5 + y

Now all we need to do is plug this equation into equation (1), therefore substituting x with y + 5:

(y +5) + 4y = 20

We then solve this like any other equation:

5y = 15

y =3

We've now found y! But it's not over just yet! Don't forget we also have to find x! For this, we just put in y's value into one of our equations and then solve it:

x + 4*3 = 20

x + 12 = 20

x = 8

And now we've also found x! Yay! To check if your answers are right, you can change the values of x and y in our equations, and you will see that we get the correct results!

There is another method to solve simultaneous equations, called elimination. In this method, we want to make the coefficient of one of the variables the same in both equations. We then substract one equation from the other, thus eliminating the variable with the same coefficient. If this seems a little confusing, let's take our two equations again: 

(1) x + 4y = 20 

(2) 2x - 2y = 10 

What we can do is multiply (1) by 2, giving us:

2x + 8y = 40

We then substract both equations:

2*(1) - (2) : 2x + 8y -(2x -2y) = 40 -10

(Note that we have - (-2y), which in the end gives us +2y)

2*(1) -(2) : 10y = 30

y = 3

Yay, we found y again! And now, we just have to slip our y into one of our equations to find that the value of x is 8. Not too bad, is it?

Jonathan W. A Level Computing tutor, A Level French tutor, GCSE Maths...

12 months ago

Answered by Jonathan, a GCSE Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Natalie B.

Degree: Biochemistry (Masters) - Oxford, St Edmund Hall University

Subjects offered: Maths, Chemistry+ 1 more


“Currently studying biochemistry at Oxford University, and ready to help you bump up your grades/catch up on missed classes or difficult topics/rekindle lost love of the sciences and make the grades you need for your next step. We can ...”

£18 /hr

George S.

Degree: Physics (Masters) - Birmingham University

Subjects offered: Maths, Physics+ 2 more

Further Mathematics

“About Me: I am just entering my first year to study Physics at University of Birmingham, I've always enjoyed anything maths based hence my A-level choices, I hope I can help you through your work whether it's GCSE or A-level and maybe...”

£24 /hr

Karolina K.

Degree: Dentistry (Bachelors) - Kings, London University

Subjects offered: Maths, Science+ 5 more

-Personal Statements-
-Medical School Preparation-

“Hi, my name is Karolina and I am a Dentistry Student from King's College London.  I understand that many students find certain aspects of the A-level syllabus challenging and having recently completed mine I believe that I know how to...”

About the author

Jonathan W.

Currently unavailable: for regular students

Degree: Computer Science and Artificial Intelligence (Bachelors) - Birmingham University

Subjects offered: Maths, French+ 1 more


“About Me: I am a Computer Science And Artificial Intelligence student at the University of Birmingham. I am really in love with my subject- which involves a lot of maths, as well as computing- and Ienjoy teaching other people about th...”

You may also like...

Posts by Jonathan

How do we solve simultaneous equations, say for example x + 4y = 20 and 2x - 2y = 10 ?

What is the difference between an array and an ArrayList in Java?

When should we use the imperfect tense or the simple past tense?

Other GCSE Maths questions

Factorise x² + 10x + 16

Solve the inequality 6y + 5 > 8

What is 3!/5! written in standard form?

If there are 20 marbles in a bag and 1/5 are red, what fraction are red if 1 more red marble is added?

View GCSE Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss