1) X - 0 = 0 - X = X

Well, X - 0 = X, while 0 - X = -X. That means X = -X and 2X = 0. The statement can only be true if X = 0. E).

2) X - Y = Y - X

This runs into the same case as 1) for the most part. Y - X = -1(X - Y), and the result is 2(X - Y) = 0. That means X - Y = 0, so X = Y. D).

3) (X - Y) - Z = X - (Y - Z)

Careful with this one. The left side is X - Y - Z, and the right side is X - Y + Z. This is easily gotten incorrect because of failing to distribute the minus sign. X - Y on both sides cancels out. They can be any number. -Z = Z runs into the same case as 1); Z = 0 is necessary. So the CLOSEST answer is A). There's a reason I say CLOSEST answer and not the CORRECT answer. Remember I said X - Y was on both sides and that X and Y can be any number? X >= Y isn't necessarily the case; if X = 5 and Y = 7, e.g., (5 - 7) - 0 = 5 - (7 - 0) ---> -2 = -2, but X < Y. So in truth NONE of the answers are right; Z = 0 independent of what X and Y are.

1. You can rewrite X - 0 as X and 0 - X as -X.

So this equation says X = -X. The only number that's true for, is 0.

2. Let's consider those possibilities one at a time.

a. X = 0. Then it becomes 0 - Y = Y - 0 or -Y = Y. We discussed this above. There's only one solution, so "Y any integer" is clearly wrong.

b. Try two integers. What is 3 - 2? What is 2 - 3? Are they the same?

c. Y = 0, so the equation becomes X - 0 = 0 - X which is the same as #1. Is "X any integer" a solution?

d. If X = Y, then X - Y = 0 and Y - X = 0.

3. Let's take them one at a time.

If you remove the parentheses, you get

X - Y - Z = X - Y + Z (a minus sign outside parentheses negates everything inside the parentheses).

a. Z = 0, it becomes X - Y = X - Y, which is obviously true for any X and Y. No restriction is needed. So this is a partial solution.

b. X = Y, it reduces to 0 - Z = 0 + Z or -Z = Z. Clearly not true for Z any integer, only 0.

c. X = 0, it becomes -Y - Z = -Y + Z. Again only true if Z = 0.

d. X = Y = Z, it becomes 0 - Z = 0 + Z. Again only true for Z = 0.