If $\frac{x}{2}+\frac{y}{3}=4$ and $\frac{2}{x}+\frac{3}{y}=1$, then what is $x+y$ equal to?

Let's simplify the given equations:
Equation 1:

$$\frac{3x+2y}{6}=4⟹3x+2y=24$$

Equation 2:

$$\frac{2y+3x}{xy}=1⟹3x+2y=xy$$

From Equation 1, we know that $3x+2y=24$. Substituting this into Equation 2 gives:

$$xy=24$$

Now we have a system:

1. $3x+2y=24$
2. $xy=24⟹y=\frac{24}{x}$

Substitute $y$ into Equation 1:

$$3x+2\left(\frac{24}{x}\right)=24$$

$$3x+\frac{48}{x}=24$$

Divide the entire equation by 3:

$$x+\frac{16}{x}=8$$

$$x^2-8x+16=0$$

$$(x-4{)}^2=0⟹x=4$$

If $x=4$, then $y=\frac{24}{4}=6$.
Therefore, $x+y=4+6=10$.