How to solve equations of this type My math teacher gave us some questions to practice for the midterm exam tomorrow, and I noticed some of them have the same wired pattern that I don't know if it has a name:
Q1. If: $x - \frac{1}{x} = 3$ then what is $x^2 + \frac{1}{x^2}$ equal to?
The answer for this question is 11, but I don't know how, I thought it should be 9, but my answer was wrong.
Q2. If: $\frac{x}{x + y} = 5$ then what is $\frac{y}{x + y}$ equal to?
The answer for this is -5, I also don't know how.
Q3. If: $x^4 + y^4 = 6   x^2  y^2 \land x\neq y$ then what is $\frac{x^2 + y^2}{x^2 - y^2}$ equal to?
I guess the answer for this was $\sqrt{2}$ but I'm not sure.
Any body can explain how to solve these questions, and questions of the same pattern?

Hint:
1: $x - \dfrac{1}{x} = 3 \implies x^2 + \dfrac{1}{x^2} - 2 = 9$
Now use, $\big(x + \dfrac{1}{x}\big)^2 = x^2 + \dfrac{1}{x^2} + 2$
2: $\dfrac{x}{x+y} = \dfrac{1}{1 + \dfrac{y}{x}}$
3: $x^4 + y^4 = 6 * x^2 * y^2 \implies \dfrac{x^2}{y^2} + \dfrac{y^2}{x^2} = 6$, solve for y/x or x/y use that to get the value of final expression
