Here is my task:
Find two solutions of system of equations:
$y'=y+3z$
$z'=y-z$,
Check (using the Wronskian) if the solutions are linearly independent. Then write a general solution, and then find solution which satisfies conditions $y(0)=-1$, $z(0)=1$.
What does "Find two solutions" mean? Why "two" and not maybe "three" or "one"?
I found the general solution using a standard technique for solving this type of systems, and I got
$$ y(x)=C_1 e^{2x}+C_2e^{-2x}\\ z(x)=\frac{1}{3}(C_1e^{2x}-3C_2e^{-2x})$$
Solving initial value problem, I got $C_1=0$ and $C_2=-1$, so particular solutions are $y(x)=-e^{-2x}$, $z(x)=e^{2x}$.
The theory tells you that a system of $n$ first order linear equations will have $n$ linearly independent solutions. One way to check this in calculations is to find $n$ solutions and then compute the Wronskian determinant for these solutions. If it is never zero, then you have linearly independent solutions.
Since you know how to find the general solution to your problem, this doesn't really concern you very much. Put another way, your question could have been posed as "find a basis for the solution set of ...; check using the Wronskian determinant that your basis is linearly independent; find a solution satisfying these initial conditions."