A fraction problem a = x + 1 x b = y + 1 y...

Jamiya Weber

Jamiya Weber

Answered

2022-06-25

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

Answer & Explanation

assumintdz

assumintdz

Expert

2022-06-26Added 22 answers

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 b
and
c 1 c 2 = a 2 + b 2 4.
Use the quadratic formula to solve for the possible values of c
Semaj Christian

Semaj Christian

Expert

2022-06-27Added 12 answers

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 + 1
Multiply 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 + 2
Combining these two lines,
a b x y = c + a x + b y 2
At this point, I'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't have to square the thing or anything messy like that.

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