Chapter 10.9, Problem 22E

To determine

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

Answer to Problem 22E

The region of solution set of system of equation $3x-y<21$ and $2x+3y>12$ is unbounded.

The coordinates of all vertex is $\left(\frac{75}{11},\frac{-6}{11}\right)$

Explanation of Solution

Given:

  $\{\begin{array}{l}2x+3y>12\hfill \\ 3x-y<21\hfill \end{array}$

Calculation:

Consider the following inequalities:

  $\{\begin{array}{l}2x+3y>12\hfill \\ 3x-y<21\hfill \end{array}$

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 $2x+3y>12$

Graph the equation of line $2x+3y=12,$ corresponding to the inequality $2x+3y>12$

  

Consider a point $(0,0),$ and test whether the inequality $2x+3y>12$ satisfy the point (0,0)

or not

Substitute $x=0$ and $y=0,$ in $2x+3y>12$

  $2x+3y>12$

  $2(0)+3(0)>12$

  $0+0>12$

  $0>12$

  $0>12$ is not true. Therefore, (0,0) does not satisfy the inequality $2x+3y>12$ and (0,0) is not a part of the graph of $2x+3y>12$

The graph of $2x+3y>12$ is the region above the line $2x+3y=12$

The graph of inequality $2x+3y>12$ is shown below in yellow color:

  

Consider the second inequality $3x-y<21$

Graph the equation of line $3x-y=21,$ corresponding to the inequality $3x-y<21$

  

Consider a point $(0,0),$ and test whether the inequality $3x-y<21$ satisfy the point (0,0) or

not.

Substitute $x=0$ and $y=0,$ in $3x-y<21$

  $3x-y<21$

  $3(0)-(0)<21$

  $\begin{array}{r}\hfill 0-0<21\\ \hfill 0<21\end{array}$

  $0<21$ is true. Therefore, (0,0) satisfies the inequality $3x-y<21$ and (0,0) is a part of the

graph of $3x-y<21$

Graph of $3x-y<21$ is region above the line $3x-y=21$

The graph of $3x-y<21$ is shown below in green color:

  

Combine the graph of $2x+3y>12$ and the graph of $3x-y<21$ .

  

The region in clay color is the intersection of the graph of $2x+3y>12,$ and $3x-y<21$

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

Solve the equation of line corresponding to $2x+3y>12$ and $3x-y<21$

  $2x+3y=12$

  $3x-y=21$

Multiply both sides of $3x-y=21$ by 3

  $3(3x)-3(y)=3(21)$

  $\begin{array}{l}9x-3y=63\\ \text{Add }2x+3y=12\text{and }9x-3y=63\\ 2x+3y+9x-3y=63+12\\ 2x+9x=75\\ 11x=75\\ x=\frac{75}{11}\end{array}$

Substitute $x=\frac{75}{11},$ in $2x+3y=12$

  $2x+3y=12$

  $2\left(\frac{75}{11}\right)+3y=12$

  $\frac{150}{11}+3y=12$

  $3y=12-\frac{150}{11}$

  $3y=12-\frac{150}{11}$

  $3y=\frac{132-150}{11}$

  $3y=\frac{-18}{11}$

  $y=\frac{-6}{11}$

The coordinates of all vertex is $\left(\frac{75}{11},\frac{-6}{11}\right)$

  

The region of solution set of system of equation $3x-y<21$ and $2x+3y>12$ is not enclosed in a circle.

Therefore, the region of solution set of system of equation $3x-y<21$ and $2x+3y>12$ is unbounded.

Conclusion:

Therefore, the region of solution set of system of equation $3x-y<21$ and $2x+3y>12$ is unbounded.

The coordinates of all vertex is $\left(\frac{75}{11},\frac{-6}{11}\right)$