This one must be pretty basic, but...
Prove that for all real numbers $x$ and $y$ there is a real number $z$ such that $x + z = y - z$
I am quite confused about how you need to prove this.
My attempt was: $$\tag1 x + z = y - z$$ $$\tag2 2z = y -x $$ $$\tag3 z = \frac{y-x}{2}$$ Denominator doesn't equal to zero, hence $z$ is defined for all $x$ and $y$.
Is there anything else I need to show?
If this question is from an analysis book, you might have to explain it in addition to solving it. Strictly speaking, it is a mathematical statement. The first part of the statement, "for all $x$ and $y$," implies that values of $x$ and $y$ have already been stipulated. You have no control over them. The second part of the statement, "there is a real number $z$," implies that you only have to find one real number $z$ that satisfies $x + z = y - z$. Solving this equation for $z$ shows that you can always find this value of $z$.