Question

If x/y +y/x=-1 , x not equal to 0 and y not equal to 0 ,find value of x^3-y^3​

Answer 1

We note that

${x}^{3} − {y}^{3} =(x−y)( {x}^{2} +xy+ {y}^{2} )$

Now, if xy+yx=−1 , then, by getting a common denominator, we find that

$\frac{ {x}^{2} + {y}^{2} }{xy} = - 1$

Multiply through by xy :

${x}^{2} + {y}^{2} =−xy$

bring all terms to the left:

${x}^{2} +xy+ {y}^{2} =0$

Hence,

${x}^{3} - {y}^{3} = (x - y)( {x}^{2} + {y}^{2} + xy) \\ (x - y)0 = 0$

Here you are done.