Expert: Ahmed Salami - 9/15/2009

Question
Hi.
Can you check the error in this problem I tried to solve it but it turned out different than the the answer in the text book
dy/dx of  x^2+y^2-2xy= x + y - 1 .
my solutions:
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2x + 2y(x) *y'(x) -(2x*y)'=1+y'(x)-0 >> (1)

2y(x) *y'(x) - (2x*y'(x)+2y(x)=1+y'(x)-2x >> (2)

2y(x)*y'(x) - 2x*y'(x) -2y(x)=1+y'(x)*(x)-2x >> (3)

2y(x)*y'(x) - 2x*y'(x)-y'(x) (x)=1-2x+2y(x) >> (4)

y'(x)(2y(x)-2x-1)=1-2x+2y(x) >> (5)

y'(x)(2(y(x)-x)-1)=1-2x+2y(x) >> (6)

y'(x)=1-2(x+y(x))/(2(y(x)-x)-1)  >> (7)
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but the answer in my text book is
2(y-x)+1 / 2(y-x)-1
I numbered my steps so you can tell me which one is incorrect

Answer
Hi Hamad,
Just go through my solution.
x^2 + y^2 - 2xy = x + y - 1
2x + 2y(y') - 2(xy' + y) = 1 + y'
2x + 2y(y') - 2xy' - 2y = 1 + y'
2y(y') - 2xy' - y' = 1 - 2x + 2y
y'(2y - 2x - 1) = 1 - 2x + 2y
y' = (1 - 2x + 2y)/(2y - 2x - 1)
  = 2y - 2x + 1 / 2y - 2x - 1
  = 2(y-x) + 1 / 2(y-x) - 1

The problem is with your line 7
It should be y'(x)=1-2(x-y(x))/(2(y(x)-x)-1)  >> (7)
i.e x-y(x) and not x+y(x)

Regards