Find the smallest positive integer solution to $\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}$.
By the tangent addition formula,
\begin{align*}
\frac{\cos 96^\circ + \sin 96^\circ}{\cos 96^\circ - \sin 96^\circ} &= \frac{1 + \tan 96^\circ}{1 - \tan 96^\circ} \\
&= \frac{\tan 45^\circ + \tan 96^\circ}{1 - \tan 45^\circ \tan 96^\circ} \\
&= \tan (45^\circ + 96^\circ) \\
&= \tan 141^\circ.
\end{align*}Thus, we seek the smallest positive integer solution to
\[\tan 19x^\circ = \tan 141^\circ.\]This means $19x - 141 = 180n$ for some integer $n,$ or $19x - 180n = 141.$  We can use the Extended Euclidean Algorithm to find the smallest positive integer solution.

Running the Euclidean Algorithm on 180 and 19, we get
\begin{align*}
180 &= 9 \cdot 19 + 9, \\
19 &= 2 \cdot 9 + 1, \\
9 &= 9 \cdot 1.
\end{align*}Then
\begin{align*}
1 &= 19 - 2 \cdot 9 \\
&= 19 - 2 \cdot (180 - 9 \cdot 19) \\
&= 19 \cdot 19 - 2 \cdot 180.
\end{align*}Multiplying both sides by 141, we get
\[2679 \cdot 19 - 282 \cdot 180 = 141.\]Note that if $(x,n)$ is a solution to $19x - 180n = 141,$ then so is $(x - 180,n + 19).$  Thus, we reduce 2679 modulo 180, to get $x = \boxed{159}.$

Alternatively, we want to solve
\[19x \equiv 141 \pmod{180}.\]Multiplying both sides by 19, we get
\[361x \equiv 2679 \pmod{180},\]which reduces to $x \equiv \boxed{159} \pmod{180}.$