Question

13: Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum value of the function.

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

Short answer

The maximum value of the function is $17$at $17$ and the minimum value of the function is $-15$ at $\left(0,5\right)$

Step-by-step solution

Step-1 – Concept of linear inequalities.

The solution of inequalities can be obtained by changing the inequalities into equations and solving the linear equations.

Step-2 – Concept of shading the region of inequality.

The shaded region is obtained by choosing a point, if the point satisfy the inequalities then the shaded region is towards the point otherwise the shaded region is away from the point.

Step-3 – Evaluation of solution.

Given the inequalities are-:

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

The equation of the respective inequalities is $y=5,y=-3$

$\begin{array}{c}4x+y=5\\ -2x+y=5\end{array}$

The points which satisfy the equation $4x+y=5$ are $(0,5)$ and$(1,1)$.

The points which satisfy the equation $-2x+y=5$ are $(0,5)$ and $(1,7)$.

Step-4 – Shading the region.

We choose $(0,0)$ point which satisfies the inequalities $5\ge y\ge -3$, $4x+y\le 5$ and $-2x+y\le 5$.

Step-5 – Plotting the graph.

So, the graph of the inequality is-:

The common shaded region is $ABC$, where $A(0,5),B(2,-3),C(-4,-3)$.

Step-6 – Evaluating the maximum and minimum value of the function.

The given function is $f(x,y)=4x-3y$at point $A$, $f(x,y)=-15$

At point $B$, $f(x,y)=4\left(2\right)-3(-3)$

$\begin{array}{l}=8+9\\ =17\end{array}$

At point$C$, $f(x,y)$$=4(-4)-3(-3)$