Consider the following simultaneous equations in $x$ and $y$...where $a$ is a real constant: $x+y+axy=a$,$x-2y-xy^2$

Question

Consider the following simultaneous equations in $x$ and $y$:
$$x+y+axy=a$$ $$x-2y-xy^2=0$$
where $a$ is a real constant. Show that these equations admit real solutions in $x$ and $y$.
I could not approach the problem.I have no idea how to solve these kind of problems. Please help!

[N.B.-I am not sure of the tag,please feel free to recommend]

Answer 1

HINT:

Eliminating one of the two unknowns, say $y$ we can form a Cubic equation in $x$

Using Complex conjugate root theorem,

an odd degree equation with real coefficients has an odd number of (at least one) real root(s)

From the first equation given, if $x$ is real so will be $y$ and vice versa