Find the dual of:

maximize [tex]z = 6x_1 - 3x_2 - 2x_3 + 5x_4[/tex]

subject to:

[tex]4x_1 + 3x_2 - 8x_3 + 7x_4 = 11[/tex]

[tex]3x_1 + 2x_2 + 7x_3 + 6x_4 \geq 23[/tex]

[tex]7x_1 + 4x_2 + 3x_3 + 2x_4 \leq 12[/tex]

[tex]x_1, x_2 \geq 0[/tex]

[tex]x_3 \leq 0[/tex]

[tex]x_4 \text{ is unrestricted}[/tex]

To find the dual of the given linear programming problem, we need to convert it to standard form and then convert the dual constraints and objective function.

To find the dual of the given problem, we need to convert it to standard form. First, we introduce slack variables for the inequality constraints and express all inequalities as equalities. Then, we introduce surplus variables for the greater-than-or-equal-to constraints and express them as equalities. Finally, we convert the dual constraints and objective function.

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Find the dual of:maximize [tex]z = 6x_1 - 3x_2 - 2x_3 + 5x_4[/tex]subject to:[tex]4x_1 + 3x_2 - 8x_3 + 7x_4 = 11[/tex][tex]3x_1 + 2x_2 + 7x_3 + 6x_4 \geq 23[/tex][tex]7x_1 + 4x_2 + 3x_3 + 2x_4 \leq 12[/tex][tex]x_1, x_2 \geq 0[/tex][tex]x_3 \leq 0[/tex][tex]x_4 \text{ is unrestricted}[/tex]

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