Question

Eliminate $x,y$ from the equations $x-y=a,x^2-y^2=b^2,x^3-y^3=c^3$.

Answer 1

$x-y=a$ ....... $\left(i\right)$

$x^2-y^2=b^2$.
$(x+y)(x-y)=b^2$....... $\left(ii\right)$

$x^3-y^3=c^3$ ....... $\left(iii\right)$
$\Rightarrow (x-y)(x^2+xy+y^2)=c^3$
$a(x+y)=b^2$ ......... [From $\left(ii\right)$ and $\left(i\right)$]
$a^2(x+y{)}^2=b^4$
$a^2\left(\right(x-y{)}^2+4xy\left)\right)=b^4$
$a^2(a^2+4xy)=b^4$
$xy=\frac{b^4-a^4}{4a^2}$

$a(x^2+xy+y^2)=c^3$ ......... [From $\left(i\right)$ and $\left(iii\right)$]
$a\left(\right(x-y{)}^2+3xy)=c^3$
By substituting the values of $x-y,xy$ in the above equation, we can eliminate the $x,y$
$a(a^2+\frac{3(b^4-a^4)}{4a^2})=c^3$
Therefore, the eliminated equation is
$a^4-4a{c}^3+3b^4=0$