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?

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