Questions: Write the e-system obtained via slack variables for the given linear programming problem. Maximize P=4x1+6x2 subject to 5x1+2x2 ≤ 12 7x1+6x2 ≤ 14 x1, x2 ≥ 0 Write the associated linear equation for the inequality 5x1+2x2 ≤ 12. Use s1 as the slack variable. 5x1+2x2+s1=12 Write the associated linear equation for the inequality 7x1+6x2 ≤ 14. Use s2 as the slack variable.

Write the e-system obtained via slack variables for the given linear programming problem. Maximize P=4x1+6x2 subject to 5x1+2x2 ≤ 12 7x1+6x2 ≤ 14 x1, x2 ≥ 0 Write the associated linear equation for the inequality 5x1+2x2 ≤ 12. Use s1 as the slack variable. 5x1+2x2+s1=12 Write the associated linear equation for the inequality 7x1+6x2 ≤ 14. Use s2 as the slack variable.
Transcript text: Write the e-system obtained via slack variables for the given linear programming problem. \[ \begin{aligned} \text { Maximize } & P=4 x_{1}+6 x_{2} \\ \text { subject to } & 5 x_{1}+2 x_{2} \leq 12 \\ & 7 x_{1}+6 x_{2} \leq 14 \\ & x_{1}, x_{2} \geq 0 \end{aligned} \] Write the associated linear equation for the inequality $5 \mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 12$. Use $\mathrm{s}_{1}$ as the slack variable. $5 x_{1}+2 x_{2}+s_{1}=12$ Write the associated linear equation for the inequality $7 \mathrm{x}_{1}+6 \mathrm{x}_{2} \leq 14$. Use $\mathrm{s}_{2}$ as the slack variable.
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Solution

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Solution Steps

Step 1: Introduction to Slack Variables

In linear programming, slack variables are introduced to convert inequalities into equalities. This is done by adding a non-negative slack variable to each inequality constraint.

Step 2: Convert the First Inequality

The first inequality given is: \[ 5x_1 + 2x_2 \leq 12 \] To convert this into an equation, we introduce a slack variable \(s_1 \geq 0\): \[ 5x_1 + 2x_2 + s_1 = 12 \]

Step 3: Convert the Second Inequality

The second inequality given is: \[ 7x_1 + 6x_2 \leq 14 \] To convert this into an equation, we introduce another slack variable \(s_2 \geq 0\): \[ 7x_1 + 6x_2 + s_2 = 14 \]

Final Answer

The system of equations obtained by introducing slack variables is: \[ \begin{aligned} 5x_1 + 2x_2 + s_1 &= 12 \\ 7x_1 + 6x_2 + s_2 &= 14 \\ x_1, x_2, s_1, s_2 &\geq 0 \end{aligned} \]

\(\boxed{\begin{aligned} 5x_1 + 2x_2 + s_1 &= 12 \\ 7x_1 + 6x_2 + s_2 &= 14 \end{aligned}}\)

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