jmc

Since the problem asks for $a+b$, we look for a way to isolate $a+b$ from the given equations.

Notice that $a+6a=7a$ and $4b+3b=7b$. This gives us the key to isolating $a+b$. We simply add the two equations together: $$\begin{array}{rl}7a+7b& =84\\ 7(a+b)& =84\\ a+b& =\frac{84}{7}\\ a+b& =\boxed{12}\end{array}$$