Solution of system of equations in prime fields In 'Algebra', Artin writes that the system of equation:

$$8x+3y = 3$$
$$2x+6y = -1$$

have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates to $42$ which is equal to $0$ (mod $2$, $3$ and $7$). 

However he mentions that there happens to be a solution $(3, 0)$ in $\mathbb{F}_7$ though the determinant is zero modulo $7$. However this solution is not evident from the non-zero determinant (invertible matrix) rule we generally look at to determine whether solutions exist or not. 

Should we always reduce the coefficient matrix elements (modulus $p$) before evaluating value of determinant to check if  a solution exists in $\mathbb{F}_p$? Is there a better way to check for existence of solutions in prime fields?
It is evident from the matrix. in the field $Z/7Z,$ the system is
$$ x+3y = 3, $$
$$ 2x+6y = 6.      $$   

Solutions are $(3,0), \; \; $  $(0,1), \; \; $  $(4,2), \; \; $   $(1,3), \; \; $   $(5,4), \; \; $  $(2,5), \; \; $  $(6,6). \; \; $
It is evident from the matrix. in the field $Z/7Z,$ the system is
$$ x+3y = 3, $$
$$ 2x+6y = 6.      $$   

Solutions are $(3,0), \; \; $  $(0,1), \; \; $  $(4,2), \; \; $   $(1,3), \; \; $   $(5,4), \; \; $  $(2,5), \; \; $  $(6,6). \; \; $