Question 1184298
The system is the same as

(1-c)x + 3y + 5z=0
5x + (1-c)y + 3z=0
3x + 5y + (1-c)z =0

Now let {{{abs(matrix(3,3,1-c, 3,5,5,1-c,3,3,5,1-c))=0}}}

===> {{{108+42c+3c^2-c^3=0}}} ===> c = 9.  (Other roots are complex.)

Therefore the system will have an infinite number of solutions if {{{highlight(c=9)}}}.