Sum of three variables given two equations Given $$x^2+y^2+z^2=121$$ 
$$x\sqrt{11} + 4y + z\sqrt{22}=77$$
Find $$ \frac{\sqrt{11} + 4 + \sqrt{22}}{x+y+z} $$  
I tried to plug in something for z at first, since x and y should have unique values for every value of z, but that didn't seem to work.
The answer is 7/11, which is clearly the second equation divided by the first but I don't understand how or why that would lead the final expression.

HINT:
$$49\cdot121-77^2=(x\sqrt{11}+4y+z\sqrt{22})^2-(x^2+y^2+z^2)(11+16+22)$$
$$\iff0=(4x-\sqrt{11}y)^2+(\sqrt{22}x-\sqrt{11}z)^2+(\sqrt{22}y-4z)^2$$
