Chapter 10.9, Problem 21E

To determine

To find: the coordinates of all the vertices and determine whether the solution set bounds or not

Answer to Problem 21E

The region of solution set of system of equation $x+y\le 4$ and $y\ge x$ is unbounded.

The coordinates of all Vertices is (2, 2)

Explanation of Solution

Given:

  $\{\begin{array}{l}x+y\le 4\hfill \\ y\ge x\hfill \end{array}$

Calculation:

Consider the following inequalities:

  $\{\begin{array}{l}x+y\le 4\hfill \\ y\ge x\hfill \end{array}$

The objective is to sketch the graph of solution set of above-mentioned inequalities.

And, then find the coordinates of all vertices and check whether the solution set is bounded or not.

Consider the first inequality $x+y\le 4$

Graph the equation of line $x+y=4,$ corresponding to the inequality $x+y\le 4$

  

Consider a point $(0,0),$ and test whether the inequality $x+y\le 4$ satisfy the point (0,0) or not.

Substitute $x=0$ and $y=0,$ in $x+y\le 4$

  $x+y\le 4$

  $0+0\le 4$

  $0\le 4$

  $0\le 4$ is true.

Therefore, (0,0) does satisfy the inequality $x+y\le 4$ and (0,0) is part of the graph of $x+y\le 4$

The graph of $x+y\le 4$ is the region below the line $x+y=4$

The graph of inequality $x+y\le 4$ is shown below in yellow color:

  

Consider the second inequality $y\ge x$

Graph the equation of line $y=x,$ corresponding to the inequality $y\ge x$

  

Consider a point $(1,0),$ and test whether the inequality $y\ge x$ satisfy the point (1, 0) or not.

Substitute $x=1$ and $y=0,$ in $y\ge x$

  $y\ge x$

  $0\ge 1$

  $0\ge 1$ is not true. Therefore, (1, 0) does not satisfies the inequality $y\ge x$ and (1, 0) is not a part of the graph of $y\ge x$

Graph of $y\ge x$ is region above the line $y=x$

The graph of $y\ge x$ is shown below in green color.

  

Combine the above graphs

  

The region in clay color is the intersection of the graph of $x+y\le 4,$ and $y\ge x$

This is the required graph of the solution set of system of inequality.

Solve the equation of line corresponding to $x+y\le 4$ and $y\ge x$

  $x+y=4$

  $y=x$

Substitute $y=x,$ in $x+y=4$

  $x+y=4$

  $x+x=4$

  $2x=4$

  $x=2$

Substitute $x=2,$ in $y=x$

  $y=x$

  $y=2$

The coordinates of all Vertices is (2, 2)

  

The region of solution set of system of equation $x+y\le 4$ and $y\ge x$ is not enclosed in a circle.

Therefore, the region of solution set of system of equation $x+y\le 4$ and $y\ge x$ is unbounded.

Conclusion:

Therefore, the region of solution set of system of equation $x+y\le 4$ and $y\ge x$ is unbounded.

The coordinates of all Vertices is (2, 2)