Q4. - 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 for this region.

$\begin{array}{l}x\ge 0\\ y\le 0\\ y\le 2x+4\\ 3x+y\le 9\\ f(x,y)=2x+y\end{array}$

Step-by-Step Solution

Answer

 Co-ordinates of the vertex of the feasible region$OABC$ are $O\left(0,0\right),A\left(3,0\right),B\left(1,6\right),C\left(0,4\right).$

The maximum and minimum value of the function $f\left(x,y\right)=2x+y$ are $8$ and $0$ respectively.

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

$\begin{array}{c}x\ge 0\\ y\le 0\\ y\le 2x+4\\ 3x+y\le 9\end{array}$

Their respective linear equations are $x=0,y=0y=2x+4,3x+y=9.$and$4x+y=16.$

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

The points, which satisfy the equation $3x+y=9$ are $\left(0,9\right)$ and $\left(3,0\right).$

Step-4 –Evaluating the shaded region

We choose$\left(0,0\right)$ to get the shaded region. The point $\left(0,0\right)$ satisfies all inequalities $x\ge 0,y\le 0,y\le 2x+4,3x+y\le 9$

Step-5 –Plotting the graph

Therefore, the graph for the inequalities is

The feasible region is$OABC$, where co-ordinates of$O,A,B,C$are$\left(0,0\right),\left(3,0\right),\left(1,6\right),\left(0,4\right)$respectively.

Step-6 –Determination of maximum and minimum value

$f(x,y)=2x+y.$

At point $O(0,0)$

$f(x,y)=0.$

At point $A(3,0)$

$\begin{array}{c}f(x,y)=2\left(3\right)+0\\ =6.\end{array}$

At point $B(1,6)$

$\begin{array}{c}f(x,y)=2\left(1\right)+6\\ =2+6\\ =8\end{array}$

At point $C(0,4)$

$\begin{array}{c}f(x,y)=2\left(0\right)+4\\ =4.\end{array}$