Answer
$x=6$
$y=-\frac{5}{6}$
Work Step by Step
If we multiply the second equation by $-1$, then we add the two equations, we can eliminate the $x$ from the system:
$\frac{1}{2}x+\frac{1}{3}y=\frac{49}{18}$
$-\frac{1}{2}x-2y=\frac{4}{3}$
$\frac{1}{3}y-2y=-\frac{5}{3}y=\frac{49}{18}-\frac{24}{18}=\frac{25}{18}$
$y=-\frac{5}{6}$
$\frac{1}{2}x+\frac{1}{3}(-\frac{5}{6})=\frac{49}{18}$
$x=\frac{54}{9}=6$
We have to check, if the solution is right. Substitute $x$ with 6 and $y$ with $-\frac{5}{6}$:
$\frac{1}{2}(6)+\frac{1}{3}(-\frac{5}{6})=\frac{49}{18}$ $\checkmark$
$\frac{1}{2}(6)+2(-\frac{5}{6})=\frac{4}{3}$ $\checkmark$
