A fraction problem a = x + 1 x b = y + 1 y c = x y + 1 x y Express c in terms of a and b

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A fraction problem  a  =  x  +      1    x      b  =  y  +      1    y      c  =  x  y  +      1          x      y      Express   c in terms of   a and   b

assumintdz · Accepted Answer

Note that   c is not uniquely specified by   a and   b, since solving   a  =  x  +      1    x   for   x yields two solutions which are reciprocals of each other, and applying   x  ↦      1    x   or   y  ↦      1    y   to   x  y  +      1          x      y       yields       x    y    +      y    x  . However, applying either substitution again gives back   x  y  +      1          x      y      , so   c can take on either of those values.Let       c    1    =  x  y  +      1          x      y       and       c    2    =      x    y    +      y    x  . One can check that      c    1    +      c    2    =  a  band      c    1        c    2    =      a    2    +      b    2    −  4.Use the quadratic formula to solve for the possible values of   c

Semaj Christian · Accepted Answer

Multiply all things by the LCD to remove fractions.  a  x  =      x    2    +  1  b  y  =      y    2    +  1  c  x  y  =      x    2        y    2    +  1Multiply the first two equations to get  a  b  x  y  =      x    2        y    2    +  1  +      x    2    +      y    2    =  c  +      x    2    +      y    2  Adding the first two equations, we get  a  x  +  b  y  =      x    2    +      y    2    +  2Combining these two lines,  a  b  x  y  =  c  +  a  x  +  b  y  −  2At this point, I&#039;d just go back and solve for   x  ,  y using the quadratic formula,  c  +  a      (                  a        ±                              a            2                    −          4                    2        )    +  b      (                  b        ±                              b            2                    −          4                    2        )    −  2  =  a  b      (                  a        ±                              a            2                    −          4                    2        )        (                  b        ±                              b            2                    −          4                    2        )  Not the most beautiful thing in the world, but at least you don&#039;t have to square the thing or anything messy like that.

A fraction problem a = x + 1 x </mfrac> <mspace linebreak="newlin

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