Answer
$x=4, y = 3\text{ or } (4, 3)$
Work Step by Step
The system of equations is given by the following two equations:
$$\begin{cases}
\dfrac{1}{2}x+\dfrac{1}{3}y=3\hspace{20pt} &(1)\\[3mm]
\dfrac{1}{4}x-\dfrac{2}{3}y=-1 \hspace{20pt} &(2)
\end{cases}$$
Multiplying equation $(1)$ by $2$ so that the coefficients of $y$ in the two equations are additive inverses gives:
$x+\dfrac{2}{3}y=6\hspace{20pt} (1)$
$\text{Add the two equations to eliminate } y:$
\[
\begin{aligned} x+\dfrac{2}{3}y&=6 \\[2mm]
\dfrac{1}{4}x-\dfrac{2}{3}y&=-1 \ \\[3mm]
\hline \dfrac{5}{4}x&=5\\[2mm]
\end{aligned}
\]
Multiply both sides by $\dfrac{4}{5}$ to obtain:
\begin{align*}
\frac{4}{5} \cdot \frac{5}{4}x&=5\cdot \frac{4}{5}\\\\
x&=4
\end{align*}
$\text{Substituting in equation (1) for }x=4$ gives:
$\dfrac{1}{2} (4)+\dfrac{1}{3}y=3$
$2+\dfrac{1}{3}y =3$
$\dfrac{1}{3}y=1$
$y = 3$
Therefore, the solution is:
$\boxed{x=4 \hspace{20pt} y = 3}$
