If $4u-5v=23$ and $2u+4v=-8$, find the value of $u+v$.
This problem involves a system of two linear equations with two unknowns: $u$ and $v$. One way to solve such a system is to use elimination, which means adding or subtracting the equations to eliminate one of the variables. To eliminate $u$, I can multiply the second equation by $-2$ and then add it to the first equation. Multiplying the second equation by $-2$ gives $-4u-8v=16$. Adding this to the first equation gives $-13v=39$. Dividing both sides by $-13$ gives $v=-3$. Now that I have the value of $v$, I can plug it into either equation to find $u$. I'll use the first equation, since it has smaller coefficients. Substituting $v=-3$ into the first equation gives $4u-5(-3)=23$. Simplifying this gives $4u+15=23$. Subtracting $15$ from both sides gives $4u=8$. Dividing both sides by $4$ gives $u=2$. Now I have both $u$ and $v$, so I can find $u+v$. Adding $u$ and $v$ gives $u+v=2+(-3)$. Simplifying this gives $u+v=-1$.

