I've learned that homogeneous systems can have either: trivial or nontrivial solutions
Trivial which means $x=0$
But I'm unsure with nontrivial. Will it always have a free variable? Example: $x = 2r$, $y = 3r$, $z = r$ where r is a real number
or is it possible to have $x=3$, $y=2$, $z=5$ as nontrivial solution? (I haven't encountered this, just curious) Although I believe this would mean that the system is not homogeneous.
Best Answer
As a basic technique a system of linear equations can be represented as an augmented matrix. When the system is homogeneous, the right hand side is all zeros and can be omitted for convenience.
When the augmented matrix is put into reduced row echelon form through a sequence of elementary row operations, the solutions are easily found: variables corresponding to leading ones in the reduced row echelon form are determined by assigning arbitrary values to any variables that do not correspond to leading ones. So if all the variables correspond to leading ones, we get only the trivial solution. But if at least one variable lacks a leading one, that variable is free and can be assigned any value (any real number if that is the field of arithmetic, so an infinite number of solutions are then possible).