According to Shilov's Linear algebra (page no 7) while solving two linear equation,
$a_{11}x_1+a_{12}x_2=b_1...(I) \\ a_{21}x_1+a_{22}x_2=b_2...(II)$
solving in the usual way we get
$x_1=\Large {\frac{b_{1}a_{22}-b_{2}a_{12}}{a_{11}a_{22}-a_{21}a_{12}} }$ and $x_2=\Large{ \frac{b_{2}a_{11}-b_{1}a_{21}}{a_{11}a_{22}-a_{21}a_{12}} }$,
when I solved the two equations (I) and (II), my solutions for $x_1$ matches with the text (as shown above) but for $x_2$ I get following
$\color{Red}{x_2=\Large{ \frac{b_{1}a_{21}-b_{2}a_{11}}{a_{21}a_{12}-a_{11}a_{22}}} }$ {multiplying eq (II) by $a_{11}/a_{21}$ and then subtracting (II) from (I)}
when I used one of the online tools to solve these two equations with values $a_{11}=2, a_{12}=3, b_1=4, a_{21}=5, a_{22}=6, b_2=7$ answer was $x_1=-1, x_2=2$ which matched with the answer from my version of the formula, what is wrong here?
Also, on second thought I feel "my" solution for $x_2$ should be wrong, as the quantity $a_{11}a_{22}$ gets a minus sign which according the "Inversion number" method is NOT correct (the principal diagonal is one of the permutations of $1,2,...,n$ when all the rows/numbers are in ascending order, so the inversion number is zero, $(-1)^0=+1$ )