Chapter 10, Problem 108RE

To determine

To graph: the given solution set and find the coordination of vertices and determine whether the solution is bounded or not.

Answer to Problem 108RE

  

The coordination of vertices is (4,-4)

The solution set is unbounded by the graph

Explanation of Solution

Given:

  $x\ge 0,y\ge 0$

  $x+2y\le 12$

  $y\le x+4$

Calculation:

Here, it is given that

  $x\ge 4$

  $x+y\ge 24$

  $x\le 2y+12$

We can write

  $x=4$

  $x+y=24$

  $x-2y=12$

Equation (1) is a line parallel to the $y$ -axis

From equation (2), we have

  $x+y=24$

  $\Rightarrow \frac{x}{24}+\frac{y}{24}=1$

This is a line which passes through (24,0) and (0,24)

From equation ( 3 ), we have

  $\frac{x}{12}+\frac{y}{-6}=1$

This is a line which passes through (12,0) and (0,-6)

From equations ( 1 ) and (2), we have

  $x=4$

  $y=20$

(4,20) is the intersection point of both lines

From equations (2) and $(3),$ we have

  $\frac{\begin{array}{l}x+y=24\\ -x-2y=12\end{array}}{3y=12}$

  $y=4$

  $x=20$

(20,4) is the intersection point of lines (2) and (3)

From equations (3) and (1), we have

  $x=4$

  $x-2y=12$

  $4-2y=12$

  $2y=-8$

  $y=-4$

(4,-4) is the intersection point of lines (1) and (3)

The solution set is unbounded by the graph

  

Conclusion:

Therefore, the coordination of vertices is (4,-4)

The solution set is unbounded by the graph