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$$\begin{cases} ax+by=0&\\ cx+dy=0& \end{cases}$$ 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?

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3 Answers 3

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You need the determinant of $\begin {pmatrix} a&b\\c&d \end {pmatrix}$ to be nonzero to get a unique solution. That determinant is $ad-bc$

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The solution is unique unless the two equations say the same (and there is in fact a single equation), which occurs if the coefficients are proportional.

Hence

$$\frac ac\ne\frac bd,$$ also written $$ad\ne bc.$$

(The second form is preferred, as it works even with zero coefficients.)

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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. $$-\dfrac{a}{b} \neq -\dfrac{c}{d} $$

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