Liner Programming variables

Question

An objective function and a system of linear inequalities representing constraints are given. Complete parts a. through c. Objective Function z = 3x - 2y Constraints {1≤x≤7 {y≥2 { x - y≥ -3 a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the value in part (b) to determine the maximum value of the objective function and the value of x and y at which the maximum occurs.

Answer 1

Hi, Titus!

I hope you have some graph paper and colored pencils for this.

A. First, you need to draw vertical lines for 1≤ x ≤ 7. One at x = 1 and another at x = 7. Shade the area between these with a colored pencil

Next, draw a horizontal line for y ≥ 2. Using a different color, shade above that.

Last, solve x - y ≥ -3 for y, (remember that dividing by a negative changes the direction of the inequality).

This should give you y ≤ x + 3. You will need to graph the line y = x + 3. Find two points by first using 0 for x (0,3), then 0 for y (-3, 0) and draw this line. Using a third color, shade below that.

B. The area that has all three colors will give you a four-sided figure. If you used graph paper, and drew this carefully, you will have the points (1,2), (7,2), (1,4) and (7, 10) at the vertices.

To find the value of z = 3x - 2y , just plug in the values of those 4 points.

z = 3(1) - 2(2) = -1 z = 3(1) -2(4) = -5

z = 3(7) -2(2) = 17 z = 3(7)-2(10) = 1

C. The largest value for z is at (7, 2), z = 17.

I hope this helps!

Best, Karen