Given $x+a=y+b=z+c+1$

Now $\left|\begin{array}{lll}x& a+y& a+x\\ y& b+y& b+y\\ z& c+y& c+z\end{array}\right|=\left|\begin{array}{lll}x& a+y& a\\ y& b+y& b\\ z& c+y& c\end{array}\right|\left(C_3\to C_3-C_1\right)$

$=\left|\begin{array}{lll}x& y& a\\ y& y& b\\ z& y& c\end{array}\right|\left(C_2\to C_2-C_3\right)$

$=y\left|\begin{array}{lll}x& 1& a\\ y& 1& b\\ z& 1& c\end{array}\right|$

$R_2\to R_2-R_1$ and $R_3\to R_3-R_1$

 $y\left|\begin{array}{ccc}x& 1& a\\ y-x& 0& b-a\\ z-x& 0& c-a\end{array}\right|=y\left|\begin{array}{ccc}x& 1& a\\ a-b& 0& -(a-b)\\ z-x& 0& c-a\end{array}\right|=y(a-b)\left|\begin{array}{ccc}x& 1& a\\ 1& 0& -1\\ z-x& 0& c-a\end{array}\right|=-y(a-b)(c-a+z-x)=y(a-b)$