What is the solution of the following system? a ⋅ e − b ⋅ d = α a ⋅ f − c ⋅ d = β b ⋅ f − c ⋅ e = γ Where the unknowns a , b , c , d , e , f are real numbers and α , β , γ are fixed real numbers.

Question

What is the solution of the following system?                    a        ⋅        e        −        b        ⋅        d                            =        α                            a        ⋅        f        −        c        ⋅        d                            =        β                            b        ⋅        f        −        c        ⋅        e                            =        γ            Where the unknowns   a  ,  b  ,  c  ,  d  ,  e  ,  f are real numbers and   α  ,  β  ,  γ are fixed real numbers.

Josie Stephenson · Accepted Answer

The resultant of the first two equations with respect to variable   a is   b  d  f  −  c  d  e  +  α  f  −  β  e. You can solve this and the third equation as linear equations in   e  ,  f (assuming   α  c  −  β  b  ≠  0).

Jaqueline Kirby · Accepted Answer

Hint: the system can actually be interpreted as      (                            a                                      b                                      c                      )    ×      (                            d                                      e                                      f                      )    =      (                            γ                                      −          β                                      α                      )  where   × is the vector cross product.

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