Which point is outside the solution region of (x+y\leq 10)?

Correct answer: ((6,7))

The inequality \(x+y\leq10\) describes all points whose coordinate sum is at most 10. To test a listed point, substitute its x- and y-coordinates and check whether the resulting inequality is true. For \((2,3)\\), the sum is 5, so it is inside the region. For \((5,5)\\), the sum is 10, so it lies on the boundary and is included. For \((0,9)\\), the sum is 9, so it is also inside.

For \((6,7)\\), the sum is 13, and \(13\leq10\) is false. Therefore this point is outside the solution region, making option C correct. The boundary line is \(x+y=10\), and because the sign includes equality, points on that line belong to the solution set.