How can we solve the linear system?

Question

How can we solve the following linear system ?

$$ax+by=0 \\ cx+dy\neq 0$$

Answer 1

This is the same to have:

ax+by=0

cx+dy=f

Using Gauss elimination or any similar, then you will get (https://en.wikipedia.org/wiki/Gaussian_elimination):

c*(ax + by) - a*(cx + dy) = - af

(cb - ad)*y = - af

y = (- af) / (cb - ad)

finally

x = (- bf) / (ad - bc)

Answer 2

The solution set of the first equation is $\{(b t,-a t) \mid t \in \mathbb{R}\}$.

Then, by replacing $x$ and $y$ in the second equation, we have : $c x + d y \not = 0 \Leftrightarrow c (b t) + d (-a t) \not = 0 \Leftrightarrow (c b - d a) t \not = 0$.

If $c b - d a = 0$, then we can't have $(c b - d a) t \not = 0$ for any $t \in \mathbb{R}$ and the system has no solution.
If $c b - d a \not = 0$, then $t$ can't be equal to $0$ and the solution set of the system is $\{(b t,-a t) \mid t \in \mathbb{R^*}\}$.