Unique solution for parametric system of two equationsThe system is: x − 2 y + z = 2 α 3 ( x y + x z + y z ) = 3 α − 4

Question

Unique solution for parametric system of two equationsThe system is:  x  −  2  y  +  z  =  2  α    3  (  x  y  +  x  z  +  y  z  )  =  3  α  −  4

Bruce Bridges · Accepted Answer

Here is a &quot;by hand&quot; method to make some progress.You have from the first equation   x  +  z  =  2  (  y  +  α  )The second equation can be rewritten as   (  x  +  z  )  y  +  x  z  =  α  −      4    3    =  2  y  (  y  +  α  )  +  x  zBoth equations are symmetric in   x  ,  z so if there is a single solution we must have   x  =  zHence   x  =  z  =  y  +  α and       x    2    =  α  −      4    3    −  2  y  (  y  +  α  )  =  (  y  +  α      )    2  Now there is a quadratic in   y namely  9      y    2    +  12  α  y  +  3      α    2    −  3  α  +  4  =  0or  (  3  y  +  2  α      )    2    =      α    2    +  3  α  −  4  =  (  α  +  4  )  (  α  −  1  )This has a single root for y only if   α  =  −  4 or   α  =  1This identifies possible values for   α on the assumption that the solution is unique. However it remains to be checked whether the solutions obtained are unique for these values of   α. The method above can be used to obtain a quadratic for   x and   z with coefficients which are functions of   y. The discriminant of this quadratic - a function of   y can be used to check for uniqueness.

Unique solution for parametric system of two equations The system is: x −

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