which one is correct of these options?

Question

for the following set of simultaneous equations: $ 1.5x-0.5y=2 $ ; $4x+2y+9z=9$ ; $ 7x+y+5z=10 $

1. the solution is unique
2. infinitely many solutions exist
3. the equations are not compatible
4. finite number of multiple solutions exist

I hope the answer will be 3, is it correct?

Answer 1

• (4) can not be true since we're talking about a linear system
• $det\begin{pmatrix}1.5&-0.5&0\\4&2&9\\7&1&5\end{pmatrix}\neq0$ thus it's (1)

Answer 2

The solution is unique since the determinant is not zero. $$|A|=|\pmatrix{1.5 &-.5 &0\\ 4& 2& 9\\ 7 &1 &5} |= -20.0$$