matrix - non trivial solution for 2 linear equations

I have two equations for 'x' and 'y'. equations are as below:

$x \:( s - d) + a x + b \: y = 0 \\ (j \: +1) \: x + y \: ( s - e) + 9x= 0 $

d,e,,a,j and b are constants I want to find solutions of s for which determinant of the matrix of coefficients of x and y becomes zero (amounts of s, for which x and y have nontrivial solutions). Could anyone do that? answers with an example will be highly appreciated.

Below is the code I tried:

CoefficientArrays[{(- d) x + b y == 0, (j + 1) x + (s - e) y + 9 x == 0}, {x, y}]


mat = Normal[%]

matt = Select[mat, # != {0, 0} &]
Eigenvalues[-matt]