College Algebra, Chapter 2, 2.2, Section 2.2, Problem 82

Test the equation $x^2 y^2 + xy = 1$ for symmetry.


$
\begin{equation}
\begin{aligned}

x^2 y^2 + xy =& 1
&& \text{Given}
\\
\\
(-x)^2 y^2 + (-x)y =& 1
&& \text{Replace $x$ to $-x$}
\\
\\
x^2 y^2 - xy =& 1
&& \text{Simplify}

\end{aligned}
\end{equation}
$


Since the resulting equation is changed, we can say that $x^2 y^2 - xy = 1$ is not symmetric to $y$-axis.

Next,


$
\begin{equation}
\begin{aligned}

x^2 (-y)^2 + x(-y) =& 1
&& \text{Replace $x$ to $-x$}
\\
\\
x^2 y^2 - xy =& 1
&& \text{Simplify}

\end{aligned}
\end{equation}
$





Since the resulting equation is changed, we can say that $x^2 y^2 - xy = 1$ is not symmetric to $x$-axis.