jmc

First, we combine the fraction on the left to give $\frac{5x-3}{x+3}=\frac{1}{x+3}$. Then, multiplying both sides by $x+3$ gets rid of the denominators and leaves $5x-3=1$. Adding 3 to both sides gives $5x=4$, so $x=\boxed{\frac{4}{5}}$.