To solve this question, let's analyze both statements and see if they provide enough information to determine how many of the numbers $x$, $y$, and $z$ are negative, given that each can be no greater than 200.

Given:

1. $x \leq 200$, $y \leq 200$, $z \leq 200$.
2. We need to determine the number of negative values among $x$, $y$, and $z$.

Analyzing Statements:

Statement 1:

$x + y + z = 400$

Since $x$, $y$, and $z$ are each no greater than 200, they could all be positive and still sum to 400. For example:

• If $x = 200$, $y = 200$, and $z = 0$, the sum would be 400, and all values are non-negative.

Thus, Statement 1 alone does not help us determine if any of $x$, $y$, or $z$ are negative.

Statement 2:

$x + y = 280$

Similarly, this statement alone does not provide any information about $z$, and $x$ and $y$ could each still be positive. For instance:

• If $x = 200$ and $y = 80$, then $x + y = 280$, with both values positive.

Again, Statement 2 alone does not tell us if any values are negative.

Combining Statements 1 and 2:

Even when combining both statements:

• We have $x + y = 280$ and $x + y + z = 400$, so $z = 400 - 280 = 120$.
• Here, $x$, $y$, and $z$ can all still be positive.

Thus, neither statement alone nor the two combined are sufficient to determine how many of $x$, $y$, or $z$ are negative.

Conclusion:

The correct answer is indeed:

• Neither Statement 1 nor Statement 2 are sufficient, either together or alone.

Would you like a further breakdown or clarification on any part of this solution?

Here are some related questions to deepen understanding:

1. What if $x + y + z = 200$ and $x + y = 100$? Could we deduce anything about negativity?
2. How would the solution change if the range of values were different, say $x, y, z \leq 100$?
3. Can we use inequalities to establish the possible values of $x, y,$ or $z$?
4. What if one of the values had to be negative—could the statements help in that scenario?
5. How does the number of equations relate to the number of unknowns in determining unique solutions?

Tip: When working with inequalities and multiple unknowns, try testing boundary values to see if they satisfy all conditions.