Question

In the xy-plane, triangular region R is bounded by the lines x=0, y=0, and 4x+3y=60. Which of the following points lie inside region R?

a. (2, 18)
b. (5, 12)
c. (10, 7)
d. (12, 3)
e. (15, 2)

Answer 1

Convert the equation of the line to the slope-intercept form:

`4x+3y=60 \\ 3y=-4x+60 \\ y=-(4)/(3)x+20`

All points which lie inside region R satisfy the system of inequalities:

`x\ \textgreater \ 0 \\ y\ \textgreater \ 0 \\ y\ \textless \ -(4)/(3)x+20`

You can see all coordinates in your choices are positive, so the first two inequalities are satisfied. We must check which points satisfy the third inequality.
Plug the values (x,y) into the inequality and check:

`(2,18) \\ 18 \ \textless \ -(4)/(3) * 2+20 \\ 18 \ \textless \ -(8)/(3)+20 \\ 18\ \textless \ -2(2)/(3)+20 \\ 18\ \textless \ 17 (1)/(3) \\ false \\ \\ (5,12) \\ 12\ \textless \ -(4)/(3) * 5+20 \\ 12\ \textless \ -(20)/(3)+20 \\ 12\ \textless \ -6(2)/(3)+20 \\ 12\ \textless \ 13 (1)/(3) \\ true`


`(10,7) \\ 7 \ \textless \ -(4)/(3) * 10+20 \\ 7\ \textless \ -(40)/(3)+20 \\ 7\ \textless \ -13 (1)/(3)+20 \\ 7\ \textless \ 6 (2)/(3) \\ false \\ \\ (12,3) \\ 3\ \textless \ -(4)/(3) * 12+20 \\ 3\ \textless \ -(48)/(3)+20 \\ 3\ \textless \ -16+20 \\ 3\ \textless \ 4 \\ true`


`(15,2) \\ 2 \ \textless \ -(4)/(3) * 15+20 \\ 2\ \textless \ -(60)/(3)+20 \\ 2\ \textless \ -20+20 \\ 2\ \textless \ 0 \\ false`

The answer is B and D.