Chapter 6, Problem 1RE

Given the linear programming problem

$\begin{array}{l}\text{Maximize }P=6x_1+2x_2\\ \text{subject to }2x_1+x_2\le 8\\ x_1+2x_2\le 10\\ x_1,x_2\ge 0\end{array}$

convert the problem constraints into a system of equations using the slack variables.

To determine

The system of equations by the use of slack variable from the problem constraint for maximizing $P=6x_1+2x_2$, subject to $2x_1+x_2\le 8,x_1+2x_2\le 10\text{ and }{x}_1,x_2\ge 0$.

Answer to Problem 1RE

The system of equations for the given linear problem is $\begin{array}{c}2x_1+x_2+s_1=8\\ x_1+2x_2+s_2=10\end{array}$.

Explanation of Solution

Consider the given constraint inequalities.

$2x_1+x_2\le 8$ …… $\left(1\right)$

$x_1+2x_2\le 10$ …… $\left(2\right)$

There are two constraint inequalities, therefore two slack variables are required.

Suppose the two slack variables are $s_1,s_2$.

Add slack variable $s_1$ to inequality $\left(1\right)$.

$2x_1+x_2+s_1=8$

Add slack variable $s_2$ to inequality $\left(2\right)$.

$x_1+2x_2+s_2=10$

Therefore, the system of equations for the given linear problem is $\begin{array}{c}2x_1+x_2+s_1=8\\ x_1+2x_2+s_2=10\end{array}$.