Let $A=\pmatrix{4&2\\ 0&1},\ b=\pmatrix{5\\ 3}$ and $A\pmatrix{x_1\\ x_2}=b$ over the field $\mathbb Z_7$.
What is $x_1$?
So we need to calculate $$x_1=\frac{\det(A_1)}{\det(A)}$$
which is $\dfrac{\begin{vmatrix} 5&2\\ 3&1 \end{vmatrix}}{\begin{vmatrix} 4&2\\ 0&1 \end{vmatrix}}=\dfrac{6}{4}$
How do I proceed from here?
$$ 4\cdot 2 \equiv 1\pmod 7. $$ Multiplying both sides by $3$ we get $$ 4\cdot 6 \equiv 3 \pmod 7. $$ So $$ 4 \div 3 = 6\text{ in }\mathbb Z_7. $$