Solve the Diophantine equation 12x + 15y = 21.
First, we can divide by the greatest common divisor of 12, 15, and 21, which is 3. This gives us the equation $4x+5y=7$. Now, we see that $5\cdot1-4\cdot1=1$, so multiplying both sides by 7 gives $(5\cdot1-4\cdot1)\cdot7=(1)\cdot7\Rightarrow 5(7)-4(7)=7$. Thus, our solution is $\boxed{(x,y)=(7,-7)}$.