Question

If u= x/(y-z), v = y/(z-x), w=z/(x-y) solve​

Answer 1

uv  + vw  + uw  =  -1  if u= x/(y-z), v = y/(z-x), w=z/(x-y)

Step-by-step explanation:

If u= x/(y-z), v = y/(z-x), w=z/(x-y)

then uv  + vw  + uw = ?

uv  + vw  + uw

= (x/(y - z))(y/(z-x))  + (y/(z-x))(z/(x-y))  +  (x/(y-z))(z/(x-y))

= xy/(y - z)(z-x)  + yz/(z-x)(x-y)  + xz/(y-z)(x-y)

= (xy(x - y)  + yz(y-z) + xz(z - x)) / ((x-y)(y-z)(z-x) )

Let solve numerator only

xy(x - y)  + yz(y-z) + xz(z - x)

= xy(x - y)  + y²z - yz² + xz² - x²z

= xy(x - y)  + z²(x - y) -z(x² - y²)

= xy(x - y)  + z²(x - y) -z(x + y)(x - y)

= (x - y) ( xy  + z² - z(x + y))

= (x - y) ( xy - zx  + z² - zy)

=(x - y) (x(y - z)  - z(y - z) )

= (x - y) (x - z)(y - z)

= (x - y )(y - z)( -(z - x))

= - (x-y)(y-z)(z-x)

=> uv  + vw  + uw  =  - (x-y)(y-z)(z-x)  / ((x-y)(y-z)(z-x) )

=> uv  + vw  + uw  =  -1

Learn More :

if u=xy+yz+zx,v=x²+y²+z² and w=x+y+z

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