Problem Solving

BTGmoderatorDC wrote: ↑

Sun Feb 14, 2021 6:19 pm

In the xy-plane, triangular region S is bounded by the lines x=0,y=0, and 3x−2y=30. Which of the following points does NOT lie inside region S?

A. (1,−1)
B. (2,−10)
C. (3,−13)
D. (5,−5)
E. (8,−2)



OA C

Solution:

We see that the triangular region is a right triangle with legs on the axes and the hypotenuse on 3x - 2y = 30. We also see that the origin is one of the vertices of the triangle and the other two vertices are the x- and y-intercepts of 3x - 2y = 30, which are 10 and -15, respectively.
We can draw the lines and see the triangular region on the coordinate plane (see above) and clearly see that (3, -13) is not inside the triangular region.

Alternate Solution:

If you don’t want to draw a graph, you can still determine the point that is not inside the triangular region by determining the “lowest” point that is still inside or on the region. Example, if x = 2, then 3(2) - 2y = 30 yields y = -12. That is, the point (2, -12) is the “lowest” point with x-coordinate of 2 since it’s on the hypotenuse 3x - 2y = 30 and any point with y-coordinate < -12 will be outside the region. As we can see, since the point (2, -10) has a y-coordinate of -10, which is not less than -12, it's not outside the region. On the other hand, if x = 3, then 3(3) - 2y = 30 yields y = -10.5. That is, the point (3, -10.5) is the “lowest” point with x-coordinate of 3 on the hypotenuse 3x - 2y = 30 and any point with y-coordinate < -10.5 will be outside the region. As we can see, since the point (3, -13) has a y-coordinate of -13, which is less than -10.5, it's outside the region.

Answer: C