Answer
The numbers are, $3\ \text{and 2}$, $3\ \text{ and }-2$, $-3\ \text{and 2}$ and $-3\ \text{ and }-2$
Work Step by Step
Let the numbers be $ x $ and $ y $ such that:
$\begin{align}
& {{x}^{2}}-{{y}^{2}}=5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{I} \right) \\
& 3{{x}^{2}}-2{{y}^{2}}=19\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \text{II} \right)
\end{align}$
By multiplying equation (I) by $-2$ and adding equation (I) and equation (II): we get,
$\begin{align}
& -2{{x}^{2}}+2{{y}^{2}}=-10 \\
& \underline{3{{x}^{2}}-2{{y}^{2}}=19} \\
& {{x}^{2}}=9 \\
& x=\pm 3 \\
\end{align}$
Substitute $x=3$ in equation (I):
$\begin{align}
& {{\left( 3 \right)}^{2}}-{{y}^{2}}=5 \\
& {{y}^{2}}=4 \\
& y=\pm 2
\end{align}$
Substitute $x=-3$ in equation (II):
$\begin{align}
& {{\left( -3 \right)}^{2}}-{{y}^{2}}=5 \\
& {{y}^{2}}=4 \\
& y=\pm 2
\end{align}$
Thus, the numbers are $3\ \text{and 2}$, $3\ \text{ and }-2$, $-3\ \text{and 2}$ or $-3\ \text{ and }-2$.
