What allow you to combine equations together and then solve for the variables in a three variable system?

Question

For instance
4x+2y+3z=12
2x-3y+5z=-7
6x-y+4z=-3

I was informed that you can combine equation 3 to equation 1 forming a new equation, same for equation 2 and 3. What really confuses me is how with the new equations, you're able to come up with a solution that fits all three of the equations you start with.

Answer 1

The additive property of equality allows this to be done.