Maximize $\ x+\frac32 y\ $ subject to...

Question

I am stuck on the following problem:

Consider the linear programming problem:

Maximize $x+\frac32 y$

subject to $$2x+3y \le 16, \\ x+4y \le18,\\ x \ge 0,y \ge0.$$

If $S$ denotes the set of all solutions of the above problem,then which of the following options is correct?

1. $S$ is empty
2. $S$ is a singleton
3. $S$ is a line segment
4. $S$ has positive area

My Try:

I have drawn the lines and see that $Z_A$(where $Z$ is $x+\frac32 y)$=Value of $Z$ at $A=8$. Similarly, $Z_B=8,Z_C=\frac{27}{4},Z_D=18$. Now, I can not decide as according to my calculation , option 2 is the right choice but the answer key says ,the answer will be option 3. Can someone enlighten me? Thanks and regards to all.

Answer 1

As you noticed it, $(x,y)$ can only be in the pink area.
Obviously, the maximum will be reach on the blue or orange edge.
So you need to calculate the value for both side and you'll find max = 8 is reached on the blue side, which is a line segment.

Answer 2

Your area is the common area of these two inequalities, i.e. the area near origin. Name that $OAPC$ where $P$ is intersection of $AB,CD$.

Also note that what you want to maximize has same equation as in first constraint. Hece, all points on $PC$ satisfy and hence...