Prove there exists a real number $x$ such that $xy=y$ for all real $y$ 
Prove: There exists a real number $x$ such that for every real number $y$, we have $xy=y.$

In class I learned that I can prove a statement by:


*

*proving the contrapositive,

*proof by contradiction,

*or proof by cases.


Can I do something along the lines of $xy=y$
divide both sides by $x$ to get $y=\frac{y}{x}$
which would then be equal to $y=\frac{y}{x}=xy$
divide everything by $y$ to get $1=\frac{1}{x}=x$
so $x=1$?

There exists a real number $x$ such that for every real number $y$, we have $xy=x$

I know I somehow have to prove $x=0$ right? But I am not sure how to go about this.

Your approach, showing that $x=1$, works, but you could make it a lot simpler by dividing by $y$ to begin with. Then you get $x=1$ as a result of your first step.
For both of these, remember that you are proving that such a value exists; if you find a specific example, you don't have to explain how you found it (unless an assignment says otherwise). You aren't trying to prove that $xy=x\Rightarrow x=0$. You're proving that there is a value of $x$ where $xy=x$ is true for all $y$. If you can find that value, then your proof is simply showing that the value fits the conditions.
As you observe in the comments, you can simply say "Let $x=0$. Then $xy=0\cdot y=0=x$."
