How do I solve a pair of simultaneous equations?

An example of a pair of simultaneous equations is as follows:
4x + 3y = 10    (1) and
2x - 3y = 14    (2)
Now to solve these equations we can simply add them and solve for x, then sub that value of x in to one of the original equations and solve for y, but how and why do we do that?
Firstly we can try to eliminate one of the unknowns from equation (1) by using equation (2), in this case we can simply add them to eliminate y. So upon adding (1) and (2), we are left with:
6x = 24    (3)
As you can see, upon adding 3y to -3y, the unknown y has been taken out of the equation all together, which now allows us to find a value for x. Now dividing both sides of (3) by 6, we are left with:
x = 4
Hooray, we've manipulated our two equations and managed to find a value for x, now how can we find y?
To find the value of y, we can simply put our newly found value of x back in to (1). So lets try that and see what we get:
4(4) + 3y = 10, which can also be written as:
16 + 3y = 10   (4)
So now we can take the 16 from both sides of (4) which gives us:
3y = -6
Now dividing through by 3 we are left with:
y = -2
Hooray! So by using the two equations we were originally given and manipulating them in a certain way, we have solved them and found the values of the unknown variables x and y.