Solution of a non-linear system

Question

What is the solution of the following system?

$$ \begin{align} a \cdot e-b \cdot d & =\alpha \\ a \cdot f-c \cdot d & =\beta \\ b \cdot f-c \cdot e & =\gamma \end{align} $$

Where the unknowns $a,b,c,d,e,f$ are real numbers and $\alpha, \beta, \gamma$ are fixed real numbers.

I tried to solve it by giving it some geometric meaning, but failed.

Answer 1

The resultant of the first two equations with respect to variable $a$ is $bdf - cde + \alpha f - \beta e$. You can solve this and the third equation as linear equations in $e,f$ (assuming $\alpha c - \beta b \ne 0$).

Answer 2

Hint: the system can actually be interpreted as $$ \begin{pmatrix}a\\b\\c\end{pmatrix}\times \begin{pmatrix}d\\e\\f\end{pmatrix}= \begin{pmatrix}\gamma\\-\beta\\\alpha\end{pmatrix} $$ where $\times$ is the vector cross product.