Chapter 10.9, Problem 40E

To determine

The given system of equation has to be solved and graphed for its solutions, co-ordinates of all vertices and tells that the set of solution is bounded or not.

  $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ y\le 4\\ 2x+y\le 8\end{array}$

Answer to Problem 40E

The set of solution is:

  $\begin{array}{l}\left(x_1,y_1\right)=\left(2,4\right)\\ \left(x_2,y_2\right)=\left(0,0\right)\\ \left(x_3,y_3\right)=\left(0,4\right)\\ \left(x_4,y_4\right)=\left(4,0\right)\end{array}$

  

All solutions, or solution set is in the shaded plane so, the solution of the given inequalities is bounded.

Explanation of Solution

Given:

  $\begin{array}{l}x\ge 0\\ y\ge 0\\ y\le 4\\ 2x+y\le 8\end{array}$

Calculations:

The given eqns. are:

  $y\le 4$.......(1)

  $2x+y\le 8$.......(2)

Use the test point $\left(0,0\right)$ to determine whether which region satisfies the inequalities equation:

Substitute $\left(0,0\right)$ in eq. (1) and (2):

For eq. (1):

  $y\le 4,0\le 4$ ; it satisfy the given equation.

For eq. (2):

  $\begin{array}{l}2x+y\le 8\\ 0\le 8\end{array}$ ;

It also satisfies the given equation.

Clearly, need to shade the region on or below the solid line which does include the point $\left(0,0\right)$ .

From given data points: $x\ge 0,y\ge 0$ :

The values of $x,y$ are non-zero and the solid lines are nothing but the boundary lines. So, the graph of the given inequality equations is:

  

Now, the determine the vertex:

  $x=0$.......(4)

  $y=0$.......(5)

  $y=4$.......(6)

  $2x+y=8$.......(7)

The first vertex is the intersection point of two lines shown by eq. (6) and (7):

Put $y=4$ in eq. (7):

  $\begin{array}{l}2x+y=8\\ 2x+4=8\\ 2x=4\\ x=2\end{array}$

So, the first vertex point is $\left(x_1,y_1\right)=\left(2,4\right)$ .

Now, $x=0,y=0$ so, the second vertex is $\left(x_2,y_2\right)=\left(0,0\right)$ .

As $x=0,y=4$ so, the third vertex is $\left(x_3,y_3\right)=\left(0,4\right)$ .

The other vertex is the value of the $x$ intercept with the line of equation $2x+y=8$ :

  $\begin{array}{l}2x+y=8\\ 2x+0=8\\ 2x=8\\ x=4\end{array}$

So, $x$ intercept is $\left(4,0\right)$ .

So, the set of solution is:

  $\begin{array}{l}\left(x_1,y_1\right)=\left(2,4\right)\\ \left(x_2,y_2\right)=\left(0,0\right)\\ \left(x_3,y_3\right)=\left(0,4\right)\\ \left(x_4,y_4\right)=\left(4,0\right)\end{array}$

  

All solutions, or solution set is in the shaded plane so, the solution of the given inequalities is bounded.

Conclusion:

Hence, the given set of equations is derived as above.