Answer
Since $\det(A)\neq 0$, then the system has a unique solution.
Work Step by Step
Let $A$ be the coefficient matrix which is given by
$$A=\left[ \begin {array}{ccccc} 1&5&3&0&0\\ 4&2&5&0&0
\\ 0&0&3&8&6\\ 2&4&0&0&-2
\\ 2&0&-1&0&0\end {array} \right]
=-8\left[ \begin {array}{ccc} 1&5&3&0 \\ 4&2&5&0
\\ 2&4&0 &-2
\\ 2&0&-1&0 \end {array} \right]=-16\left[ \begin {array}{ccc} 1&5&3 \\ 4&2&5
\\ 2&0&-1 \end {array} \right]=-16\left[ \begin {array}{ccc} 1&5&3 \\ 0&-18&-7
\\ 0&-10&-1 \end {array} \right]
.$$
One can calculate $\det(A)$ as follows
$$\det(A)=-16((-18)(-7)-(-7)(-10))=-896.$$
Since $\det(A)\neq 0$, then the system has a unique solution.