Question

2. For which value(s) of a does the system of equations
(a-3)x+y=0
x+(a-3)y=0
have non-trivial solutions

Answer 1

This is a homogeneous equation of which the right-hand side is populated with zeroes.

For a homogeneous equation to have non-trivial roots, the determinant of the left-hand side must be zero.
So expand the determinant in terms of a, equate to zero and solve for a.

Since the resulting equation is a quadratic, you should expect to have two roots. In this case, the two roots are distinct.