Q14. - Algebra 2

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.

$\begin{array}{l}x\ge -10\\ 1\ge y\ge -6\\ 3x+4y\le -8\\ 2y\ge x-10\\ f(x,y)=2x+y\end{array}$

Step-by-Step Solution

Answer

The maximum value of the function is $1$at $(2.4,-3.8).$ And the minimum value of the function is $-26$at $(-10,-6).$

Step-1 –Concept of solution 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 satisfies the inequality then the shaded region is towards the point otherwise the shaded region is away from the point.

Step-3–Evaluation of solution

Given inequalities are

$\begin{array}{l}x\ge -10\\ 1\ge y\ge -6\\ 3x+4y\le -8\\ 2y\ge x-10\end{array}$

The linear equations of the respective inequalities are

$\begin{array}{l}x=-10\\ y=1,y=-6\\ 3x+4y=-8\\ 2y=x-10\end{array}$

The points, which satisfies the equation $3x+4y=-8$ are $(0,-2)$and$(-4,1)$.

The points, which satisfies the equation $2y=x-10$ are $(0,-5)$and$(10,0).$

Step-4–Shading the region

We choose the point $(0,0).$

The point$(0,0)$ satisfies the inequalities $x\ge -10,1\ge y\ge -6,2y\ge x-10$but it does not satisfies the inequality$3x+4y\le -8.$

Step-5–Plotting the graph

Therefore, the graph of the inequalities is

The common shaded region of the inequalities is$ABCDE$, where $A\left(-10,1\right),B\left(-10,-6\right),C\left(-2,-6\right),D\left(2.4,-3.8\right),E\left(-4,1\right).$

Step-6–Evaluating the maximum and minimum value

The given function is $f\left(x,y\right)=2x+y$at point$A\left(-10,1\right)$.

$\begin{array}{c}f\left(x,y\right)=-20+1\\ =-19.\end{array}$

At point$B\left(-10,-6\right)$

$\begin{array}{c}f\left(x,y\right)=-20-6\\ =-26\end{array}$

At point $C\left(-2,-6\right)$

$\begin{array}{c}f\left(x,y\right)=-4-6\\ =-10.\end{array}$

At point $D\left(2.4,-3.8\right)$

$\begin{array}{c}f\left(x,y\right)=4.8-3.8\\ =1.\end{array}$

At point $E\left(-4,-1\right)$

$\begin{array}{c}f\left(x,y\right)=-8+1\\ =-7.\end{array}$