Solving linear equation with three variables

Question

1)Do equations with 3 variables require at least 3 equation for us to solve without any dependent variable?
2)if two equations of such kind equate to zero for eg: a-b+2c=0 and 3a+b+c=0. Then using the matrice ,there is a method to solve them...explain that method.is there a proof or is it related to Gaussian method which is out of my scope?

Answer 1

1)For linear equations over the reals, yes, you need as many equations as variables. In the usual case you will have a unique solution. For equations over the naturals, you can sometimes get by with less. Also consider $x^2+y^2+z^2=0$ That looks like one equation, but there is only one solution in the reals for three variables.

2)The matrix method works fine with two equations with no constant. You may have equations that are inconsistent or duplicative, but that can happen with a constant as well.