Solving 3 simultaneous equations It's been a while since I've had to do a simultaneous equation and I'm rusty on a few of the particulars. 
Say for example I have the following equations:
x + y       = 7
2x + y + 3z = 32
    2y + z  = 13

I know that I need to combine the above 3 equations into 2 other equations, for example, if I combine (counting down) 1 + 2 I'd get
3x + 2y + 3z = 32

And combing 2 with 3 I'd get 
2x + 3y + 4z = 45

Which is fine, and I understand. It's the next steps I have trouble understanding. A lot of the examples I've been looking at have a value for each of the x, y z. Looking at this site here I'm not sure what is going on step 2. 
I can see that they are multiplying one line by 2. Is that something you always do? Like, with simultaneous equations do you always multiple one of the equations by 2? If not, how do you determine which number to use? 
My understanding of simultaneous equations is extremely limited. 

Multiply equation (1) by 2 and subtract it from equation (2).
$2x + y + 3z - 2x - 2y = 32 - 14$
$-y +3z = 18$ ......(4)
Now solve equation (3) and (4) to find y and z.
Multiply equation (4) by 2 and add with equation (3).
$2y + z -2y + 6z = 13 - 36$
$7z = -23$
$z = \frac {-23}{7}$
Then from equation (1) find x.

Way to solve.

Try to eliminate one variable and you have two new equations with remaining two variables.
Then solve these two new equations to find both variable.
And when you put value of these 2 variables to find 3rd.
