Question

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region.

$\begin{array}{l}x\le 5\\ y\le 0\\ 2y\le +7\\ y\le x-4\\ f(x,y)=4x-3y\end{array}$

Short answer

Co-ordinates of the vertex of the feasible region$ABCD$are

$A\left(1,3\right),B\left(5,1\right),C\left(5,6\right),D\left(-13,-3\right).$

The maximum and minimum value of the function $f\left(x,y\right)=4x-3y$are $17$and$-43$respectively.

Step-by-step solution

Step-1 –Concept of solving the linear inequalities

To solve the inequalities we convert the inequalities into linear equations and find the solutions of the equations to obtain the graph.

Step-2 –Concept of shading the region

For shading the region, we choose a point. If the point satisfies the inequalities then the shaded region is towards the point otherwise, the shaded region is away from the point.

Step-3 –Solving the inequalities

Given inequalities are

$\begin{array}{c}x\le 5\\ y\ge -3\\ 2y\le x+7\\ y\ge x-4\end{array}$

Their respective linear equations are$x=5,y=-3,2y=x+7,y=x-7.$

The points which satisfy the equation $2y=x+7$are $\left(-7,0\right)$and$\left(1,4\right)$

The points, which satisfy the equation $y=x-4$are$\left(0,-4\right)$and$\left(4,0\right)$.

Step-4 –Evaluating the shaded region

We choose$\left(0,0\right)$to get the shaded region. The point $\left(0,0\right)$ satisfies the inequalities $x\le 5,y\ge -3,2y\le x+7,y\ge x-4.$

Step-5 –Plotting the graph

Therefore, the graph for the inequalities is

The feasible region is$ABCD$, where co-ordinates of$A,B,C,D$are$\left(1,-3\right),\left(5,1\right),\left(5,6\right),\left(-13,-3\right)$respectively.

Step-6 –Determination of maximum and minimum value

$f(x,y)=4x-3y.$

At point $A(1,-3)$

$\begin{array}{c}f(x,y)=4-3(-3)\\ =4+9\\ =13\end{array}$

At point $B(5,1)$

$\begin{array}{c}f(x,y)=4\left(5\right)-3\\ =20-3\\ =17.\end{array}$

At point $C(5,6)$

$\begin{array}{c}f(x,y)=4\left(5\right)-3\left(6\right)\\ =20-18\\ =2.\end{array}$

At point $D(-13,-3)$

$\begin{array}{c}f(x,y)=4(-13)-3(-3)\\ =-52+9\\ =-43.\end{array}$