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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) = .

Respuesta :

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Answer:

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

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