High School

Maximize [tex]Z = 10x_1 - 4x_2 + 7x_3[/tex], subject to the following constraints:

1. [tex]3x_1 - x_2 + 2x_3 \leq 25[/tex]
2. [tex]x_1 - 2x_2 + 3x_3 \leq 25[/tex]
3. [tex]5x_1 + x_2 + 2x_3 \leq 40[/tex]
4. [tex]x_1 + x_2 + x_3 \leq 90[/tex]
5. [tex]2x_1 - x_2 + x_3 \leq 20[/tex]

Answer :

Final answer:

To maximize the given function subject to multiple constraints, you can transform it into an unconstrained problem and apply the Lagrangean function method.

Explanation:

Maximize Z=10x1 - 4x2 + 7x3, subject to:

  1. 3x1 - x2 + 2x3 ≤ 25
  2. x1 - 2x2 + 3x3 ≤ 25
  3. 5x1 + x2 + 2x3 ≤ 40
  4. x1 + x2 + x3 ≤ 90
  5. 2x1 - x2 + x3 ≤ 20

To solve this problem, you first transform it into an unconstrained one and then apply the Lagrangean function to find the extremum value of Z.

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