Use the technique developed in this section to solve the minimization problem. Minimize C = - 3x - 2y - z subject to - x + 2y - z ≤ 20 x - 2y + 2z ≤ 25 2x + 4y - 3z ≤ 30 x ≥ 0, y ≥ 0, z ≥ 0 The minimum is C = at (x, y, z) =.

C = - 145, (35/4, 295/8, 45)

Step-by-step explanation:

Use Gaussian elimination to find the values of x, y and z

Eq 1: - x+2y-z=20

Eq 2: x-2y+2z=25

Eq 3: 2x+4y-3z=30

Multily Eq1 by 1 and add to Eq 2

Eq 1: (-x+2y-z=20) * 1

Eq 2: x-2y+2z=25

Eq 3: 2x+4y-3z=30

⇒ Eq1: - x+2y-z=20

Eq2: z = 45

Eq 3: 2x+4y-3z=30

Multiply Eq 1 by 2 and then add to Eq 3

Eq1: (-x+2y-z=20) * 2

Eq2: z = 45

Eq3: 2x+4y-3z=30

⇒ Eq1: - x+2y-z=20

Eq2: z = 45

Eq3: 8y-5z = 70

swap Eq 2 and Eq 3

Eq 1: - x+2y-z=20

Eq 3: 8y-5z = 70

Eq 2: z = 45

Solve Eq 2 for z

Z=45

solve Eq Eq 3 for y.

y = 295/8

Using the value z=45 and y = 295/8, substitue in Eq 1 to get value of x

x = 35/4

Substitue values of x, y and z in C = - 3x-2y-z to get minimum value of C

C = - 145