jmc

From the first equation, we can see that $y=7-x.$ Substituting for $y$ in the second equation, we obtain $3x+(7-x)=45,$ so $x=19.$ We can then find that $y=-12.$ Thus, $$x^2-y^2=19^2-(-12{)}^2=\boxed{217}.$$- OR -

Note that $$x^2-y^2=(x+y)(x-y).$$If we subtract two times the first equation from the second equation, we obtain $x-y=31.$ We can then substitute for $x+y$ and $x-y$ to obtain $7\cdot 31=\boxed{217}.$