Use Slack Variables to Rewrite This Problem So That All $\mathrm { MAX } : \quad 2 \mathbf { X } _ { 1 } + 7 \mathbf { X } _ { \mathbf { 2 } }$

question 20

Essay

Use slack variables to rewrite this problem so that all its constraints are equality constraints.
$\mathrm { MAX } : \quad 2 \mathbf { X } _ { 1 } + 7 \mathbf { X } _ { \mathbf { 2 } }$
Subject to:
$$\begin{array} { l } \mathbf { 5 } \mathbf { X } _ { 1 } + \mathbf { 9 } \mathbf { X } _ { \mathbf { 2 } } \leq 90 \\\mathbf { 9 } \mathbf { X } _ { 1 } + \mathbf { 8 } \mathbf { X } _ { \mathbf { 2 } } \leq 144 \\\mathbf { X } _ { \mathbf { 2 } } \leq \mathbf { 8 } \\\mathbf { X } _ { 1 } \mathbf { X } _ { \mathbf { 2 } } \geq 0\end{array}$$

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Definitions:

Producer Surplus

The difference between what producers are willing to sell a good for and the actual market price they receive, representing a measure of producer welfare.

Average Cost

The total cost divided by the quantity produced, indicating the cost per unit of output.

Competitive Industry

An industry characterized by a large number of firms competing against each other, leading to innovation, varied product choices, and reasonable prices for consumers.

Marginal Cost

The cost associated with producing one more unit of a product, reflecting how production costs change with output levels.