(1,1) certainly is correct.
If you have a graphing calculator that will allow you to graph implicit functions, you will see that there is another solution when x is approximately 3 and to get it algebraically you will have to substitute.
I will work on this problem and if I can get a solution, I will get back to you.
I set up my graphing calculator incorrectly. There is another solution at x=1/4 and y=3/2.
I am going to try something I have not done before:
OK, sorry Wyzant will not let me copy the solution into this answer box and I cannot type the derivation by hand because it is much too messy. Suffice to say that if you are VERY careful with your algebra, the substitution x=(5-3y)/2 will get you both solutions via the quadratic formula. If you are careful, you will get to the quadratic equation 90y2-225y+135=0 and the quadratic formula will give you the correct solutions. I am sorry that I cannot give you the full solution, but maybe with that much of a leg up, you will be able to derive it.
It is probably worth mentioning that the graph of the 2nd degree implicit equation is an ellipse with axes at an angle to the axes of co-ordinates.
