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. 

You already know, that you can add multiple euqations, to get a *new Equation. 
I put new here, since it actually does not have any new information. 
However, instead of adding the first equation to the second, you could do something else. You could substract the second equation from the double of the first. 
This will eliminate the $x$-part in the new equation, which is a step in the right direction. 
You can continue on with more rows, more collumns, just make sure to eliminate some variables on the way. 
This is called the Gauß-Algorithm for solving linear systems. 
