# unique solution to system of two equations

$$\begin{cases} ax+by=0&\\ cx+dy=0& \end{cases}$$ How do the coefficients a, b, c and d have to be chosen for this system of equations to have one unique solution $$\begin{cases} x=0&\\ y=0& \end{cases}$$? Or what kind of conditions does one have to give to those coefficients for this system to have the unique solution?

You need the determinant of $$\begin {pmatrix} a&b\\c&d \end {pmatrix}$$ to be nonzero to get a unique solution. That determinant is $$ad-bc$$
$$\frac ac\ne\frac bd,$$ also written $$ad\ne bc.$$
Both $$ax+by = 0$$ and $$cx+dy=0$$ are straight lines passing through origin, $$(0,0)$$. For unique solution, these two shouldn't be coincident; their slopes should be distinct. $$-\dfrac{a}{b} \neq -\dfrac{c}{d}$$