Chapter 10, Problem 107RE

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 107RE

  

The coordination of vertices is $x=\frac{4}{3},y=\frac{16}{3},\left(\frac{4}{3},\frac{16}{3}\right)$

The solution set is bounded by the graph

Explanation of Solution

Given:

  $x\ge 0,y\ge 0$

  $x+2y\le 12$

  $y\le x+4$

Calculation:

It is given that,

  $x\ge 0,y\ge 0$

  $x+2y\le 12$

  $y\le x+4$

Here, it is given that $x\ge 0,y\ge 0$

  $x+2y\le 12$

  $y\le x+4$

We take

  $x+2y=12$

  $y=x+4$

  $x+2y=12\dots \dots \dots \dots .\left(1\right)$

  $x-y=-4\dots \dots \dots \dots .\left(2\right)$

From equation ( 1 ), we get

  $x+2y=12$

  $\frac{x}{2}+\frac{y}{6}=1$

This passes through (12,0) and (0,6)

From equation (2), we get

  $x-y=-4$

  $\frac{x}{-4}+\frac{y}{4}=1$

This line passes through (-4,0) and (0,4)

  $x+2y=12$

  $-x\pm y=\pm 4$

  $3y=16$

  $y=\frac{16}{3}$

  $x-\frac{16}{3}=-4$

  $x=\frac{16}{3}-4$

  $x=\frac{4}{3}$

  $x=\frac{4}{3},y=\frac{16}{3},\left(\frac{4}{3},\frac{16}{3}\right)$ are the intersection points of lines (1) and (2)

The solution set is bounded by the graph

  Conclusion:

Therefore, the coordination of vertices is $x=\frac{4}{3},y=\frac{16}{3},\left(\frac{4}{3},\frac{16}{3}\right)$

The solution set is bounded by the graph