Math Problem Statement

Express the problem as a system of equations using slack variables. (Use s1, s2, ... for the slack variables.) Maximize z = 2x1 + 4x2 + 5x3 + 9x4, subject to: x1 + 5x2 + 7x3 + x4 ≤ 82, 3x1 + 6x2 + 12x3 ≤ 50, 2x1 + 15x3 + 19x4 ≤ 240, and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0.

Solution

To convert the given constraints into equations using slack variables, we will introduce slack variables s1s_1, s2s_2, and s3s_3 to transform each inequality constraint into an equality.

The given problem is:

Maximize z=2x1+4x2+5x3+9x4z = 2x_1 + 4x_2 + 5x_3 + 9x_4, subject to:

x1+5x2+7x3+x482x_1 + 5x_2 + 7x_3 + x_4 \leq 82 3x1+6x2+12x3503x_1 + 6x_2 + 12x_3 \leq 50 2x1+15x3+19x42402x_1 + 15x_3 + 19x_4 \leq 240 x10,x20,x30,x40x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_4 \geq 0

Step-by-Step Solution

  1. First Constraint: x1+5x2+7x3+x4+s1=82x_1 + 5x_2 + 7x_3 + x_4 + s_1 = 82

  2. Second Constraint: 3x1+6x2+12x3+s2=503x_1 + 6x_2 + 12x_3 + s_2 = 50

  3. Third Constraint: 2x1+15x3+19x4+s3=2402x_1 + 15x_3 + 19x_4 + s_3 = 240

Objective Function

The objective function remains:

z=2x1+4x2+5x3+9x4z = 2x_1 + 4x_2 + 5x_3 + 9x_4

Filling the Answers

Based on this, you can fill the blanks as follows:

  • First equation: x1+5x2+7x3+x4+s1=82x_1 + 5x_2 + 7x_3 + x_4 + s_1 = 82
  • Second equation: 3x1+6x2+12x3+s2=503x_1 + 6x_2 + 12x_3 + s_2 = 50
  • Third equation: 2x1+15x3+19x4+s3=2402x_1 + 15x_3 + 19x_4 + s_3 = 240
  • Objective equation: z=2x1+4x2+5x3+9x4z = 2x_1 + 4x_2 + 5x_3 + 9x_4

Would you like a deeper explanation of each step, or have any specific questions about this process?

Here are some related questions to deepen understanding:

  1. How do slack variables affect the feasible region in linear programming?
  2. What is the purpose of converting inequalities to equations in linear programming?
  3. How can we determine if a solution is optimal in this linear programming model?
  4. What methods are commonly used to solve maximization problems in linear programming?
  5. How would adding more constraints or variables affect the complexity of the solution?

Tip: When introducing slack variables, each inequality constraint gets its unique slack variable, which transforms it into an equality suitable for simplex or tableau methods in linear programming.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Slack Variables
Optimization

Formulas

Objective function: z = 2x1 + 4x2 + 5x3 + 9x4
Slack variable equations to convert inequalities to equalities

Theorems

Fundamental Theorem of Linear Programming

Suitable Grade Level

Undergraduate