<s>[INST] Finding the value of $x-y$. Given two equations \begin{align} & x^4+y^4=\dfrac {-7}{9} \\ & x^3-y^3=3. \end{align} From these equations find the value of $(x-y).$
I have just factorize $x^3-y^3=(x-y)(x^2+xy+y^2)=3$. But don't know how to proceed from here. One way to find the values of $x$ and $y$ and then compute possible values of $x-y$. But that is a too long way. I think there is any simple way to find the value of $x-y$. Please help me to solve this.
 [/INST] Here is a hint to get you started:
$$\frac{-7}9= (x^2+y^2)^2-2(xy)^2$$$$3=(x-y)(x^2+y^2+xy)=(x-y)^3+3(xy)(x-y)$$ If we substitute $a=x^2+y^2$ and $b=xy$ and $c=x-y$, we get $$\frac{-7}9=a^2-2b^2$$$$3=c\cdot(a+b)=c^3+3bc$$
Now, you have three equations and three variables. Can you figure it out?
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