If a solution region has (x\ge 0), (y\ge 0), and (x+y\le 0), what is the solution region?

Correct answer: only ((0,0))

The direct answer is option B: only \((0,0)\). Since \(x\ge0\) and \(y\ge0\), both variables are non-negative. Their sum is therefore also non-negative. The third condition says \(x+y\le0\). A non-negative sum can be at most zero only when it is exactly zero. For two non-negative numbers, \(x+y=0\) is possible only if \(x=0\) and \(y=0\). Thus the solution region contains just the origin. Option A is wrong because the whole first quadrant contains points such as \((1,1)\), whose sum is 2 and does not satisfy the third condition. Option B is correct. Option C is wrong because the origin satisfies every inequality. Option D is wrong because the entire line \(x+y=0\) includes points with negative coordinates, which violate \(x\ge0\) or \(y\ge0\). Exam cue: non-negative quantities adding to a non-positive number must all be zero.