Finding a basis for a subspace consisting of all solutions to two different equations (check)

Question

Find a basis for the subspace of R4 consisting of all solutions to the equations $2x_1 −x_2 +x_3 −2x_4 =0$
$2x_1 + x_3 = 0$

Started by just subtracting eq2 from eq1, giving: $-x_2-2x_4$$=0$

set $x_4$=k
so $x_2=-2k$
Basis is: $(0,-2,0,1)$

Is there a mistake somewhere?

Answer 1

Consider that $2x_1+x_3=0$, now set $x_1=t$ and $x_3=-2t$ so $(1,0,-2,0)$ is also a basis. So the solutions are $$Span\{ (0,-2,0,1),(1,0,-2,0)\}$$