# Exercises 1-24, solve the given LP problem. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. [ HINT: See Example 1.] Maximize p = x + y Subject to x + 2 y ≤ 9 2 x + y ≤ 9 x ≥ 0 , y ≥ 0 .

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337274203

Chapter
Section

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
Publisher: Cengage Learning
ISBN: 9781337274203
Chapter 6.2, Problem 1E
Textbook Problem
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## Exercises 1-24, solve the given LP problem. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. [HINT: See Example 1.]Maximize p = x + y Subject to x + 2 y ≤ 9 2 x + y ≤ 9 x ≥ 0 ,     y ≥ 0 .

To determine

To calculate: The optimal solution of given linear programming problem to maximize p=x+y subject to the given following constraints 2x+y9, x+2y9 and x0,y0.

### Explanation of Solution

Given Information:

Maximize p=x+y subject to the given following constraints 2x+y9, x+2y9 and x0,y0.

Formula used:

Following the steps to find the optimal solution of linear programming problem:

1) Draw a rectangle large enough that all the corner points are inside the rectangle.

2) Shade the outside of the rectangle so as to define a new bounded feasible region and locate the new corner points.

3) Obtain the optimal solutions using bounded feasible region.

4) If the optimal solution occurs at the one of the corner points, then LP problem has that corner point as optimal solution otherwise it has no optimal solution.

Calculation:

Consider the first constraint equation,

x+2y=9

Isolate the variable y, by subtracting x from both sides,

xx+2y=9x2y=9x

Isolate the variable y, by dividing 2 on both sides,

2y2=(9x)2y=(9x)2

To plot the graph of the function, find the ordered pairs of the equation by substituting different values of x.

For x=0, substitute 0 for x,

y=(90)2=92=4.5

For x=1, substitute 1 for x,

y=(91)2=82=4

For x=3, substitute 3 for x,

y=(93)2=62=3

Construct a table for the ordered pairs,

 x 0 1 3 y 4.5 4 3

Test the first inequality by (0,0) test coordinates.

x+2y?90+20?90?9

The inequality is satisfied. If the test point satisfies the inequality then the set of solutions is the line plus the entire region on the same side of the line as the test coordinate (0,0).

Consider the second constraint equation.

2x+y=9

Isolate the variable y, by subtracting 2x from both sides,

2x2x+y=92xy=92x

To plot the graph of the function, find the ordered pairs of the second constraint equation by substituting different values of x.

For x=0, substitute 0 for x,

y=920=90=9

For x=1, substitute 1 for x,

y=921=92=7

For x=2, substitute 2 for x,

y=922=94=5

Construct a table for the ordered pairs of second constraint equation,

 x 0 1 2 y 9 7 5

Test the second inequality by (0,0) test coordinates

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