Question

For Exercises $5-8,(\text { a ) determine the number of slack variables }$ needed, (b) name them, and (c) use slack variables to convert each constraint into a linear equation. $$ \begin{array}{ll}{\text { Maximize }} & {z=1.2 x_{1}+3.5 x_{2}} \\ {\text { subject to: }} & {2.4 x_{1}+1.5 x_{2} \leq 10} \\ {} & {1.7 x_{1}+1.9 x_{2} \leq 15} \\ {\text { with }} & {x_{1} \geq 0, \quad x_{2} \geq 0}\end{array} $$

Name: Problem 6, Chapter 4: Finite Mathematics and Calculus with Applications
Uploaded: 2020-05-21T04:02:00-08:00
Duration: 42 s
Channel: Ernest Castorena
Description: For Exercises 5-8,( a ) determine the number of slack variables needed, (b) name them, and (c) use slack variables to convert each constraint into a linear equation. Maximize z=1.2 x1+3.5 x2 subject to: 2.4 x1+1.5 x2≤ 10 1.7 x1+1.9 x2≤ 15 with x1≥ 0, x2≥ 0

   For Exercises $5-8,(\text { a ) determine the number of slack variables }$ needed, (b) name them, and (c) use slack variables to convert each constraint into a linear equation. 
$$
\begin{array}{ll}{\text { Maximize }} & {z=1.2 x_{1}+3.5 x_{2}} \\ {\text { subject to: }} & {2.4 x_{1}+1.5 x_{2} \leq 10} \\ {} & {1.7 x_{1}+1.9 x_{2} \leq 15} \\ {\text { with }} & {x_{1} \geq 0, \quad x_{2} \geq 0}\end{array}
$$

Finite Mathematics and Calculus with Applications

Margaret L. Lial,… 9th Edition

Chapter 4, Problem 6

Instant Answer

Solved by Expert Ernest Castorena

05/21/2020

Step 1

In this case, we have two inequalities, so we need two slack variables. Show more…

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For Exercises $5-8,(\text { a ) determine the number of slack variables }$ needed, (b) name them, and (c) use slack variables to convert each constraint into a linear equation. $$ \begin{array}{ll}{\text { Maximize }} & {z=1.2 x_{1}+3.5 x_{2}} \\ {\text { subject to: }} & {2.4 x_{1}+1.5 x_{2} \leq 10} \\ {} & {1.7 x_{1}+1.9 x_{2} \leq 15} \\ {\text { with }} & {x_{1} \geq 0, \quad x_{2} \geq 0}\end{array} $$

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Key Concepts

Slack Variables

Slack variables are additional variables introduced to convert inequality constraints into equality constraints in a linear programming problem. They represent the unused portion or 'slack' in a constraint, and they are required to be nonnegative. By adding a slack variable to a ‘less than or equal to’ inequality, the constraint becomes an equality, which is often necessary for certain solution methods like the simplex algorithm.

Standard Form Conversion

Converting a linear programming problem into its standard form typically involves rewriting all inequality constraints as equality constraints, ensuring all variables (including added slack variables) are nonnegative, and representing the objective function in a standard maximization or minimization form. This process is critical as many optimization algorithms assume the linear program is given in this standardized representation.

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