Two ways to write a general solution of a system of linear equations.

Question

When we solve a system of linear equations in $n$ variables by Gauss elimination, there are two ways to write the general solution:

1. As one $n$-tuple depending on the free variables,
2. As a linear combination of specific vectors, with free variables as coefficients, to which a fixed vector is added.

For example:

1. $(2z+3t+4, 5z+6t+7, z+8, t+9)$,
2. $z(2,5,1,0)+t(3,6,0,1)+(4,7,8,9)$.

Are there any standard names to these two ways to write the solution?

Answer 1

For the second one you already mention "linear combination" in the description but that's what I thought it was actually called. Not sure if the first one has any special name except maybe "solution vector" or something.