How can I find x and y?

Many students ask how they can they find 2 unknowns,given 2 simple equations. I'll try to focus in this answer on giving students a tool which they can use to solve ANY 2 simple equations with one solution.Let's take an example:
5x+y=22 and 3x+4y=20
In short,the plan is to:
1.) Write y in terms of x from the first equation
2.) Substitute it in the second one, so that we will get only an equation in function of x
3.)Then, find x from it.
4.)Now, we can substitute x's value in the first equation and find y.
Concrete:
1.) Write y in terms of x from the first equation
From 5x+y=22 we get that y=22-5x.
2.) Substitute it in the second one, so that we will get only an equation in function of x
Substituing y in the second equation gives :
3x+4*(22-5x)=20.
3.)Then, find x from it. 
We now rearrange it ,so:
3x+88-20x=20(we opened the parenthesis)
Therefore:
88-17x=20(we gave x as common factor and had x*(3-20) which is -17x)
Therefore by adding 17x and subtracting 20 we get :
68=17x.
By dividing the equation with 17 we have x=4.
4.)Now, we can substitute x's value in the first equation and find y. 
So we substitute it in the first equation,so 5x+y=22 gives 54+y=22,so y=2. 
The beauty of this method stays in the fact that it can be used to solve any problem like that.
Now,with some practice,you should be able to find the solution of a similar problem. Here are some exercises which you could use to practice some more :
1.) 3x+7y=10 and x+5y=6
2.) x+y=9 and 3x+4x=32
3.) x+y=6 and x+5y= 26
4.) 4y=28 and 2x+y=9
5.) 6x-2y=72 and x+2y=12
I would finally recommend not to memorise the steps of this method,but to understand them. Good luck !
Solutions:
1.) x=1 and y=1
2.) x=4 and y=5
3.) x=1 and y=5
4.) x=1 and y=7
5.) x=12 and y=0