Questions: The partially completed table below shows the ten basic solutions to the following e-system. x1+x2+s1 =28 2 x1+x2+s2=30 4 x1+x2 +s3=40 In basic solution (J), which variables are nonbasic?

Solution Steps

To determine the nonbasic variables in basic solution (J), we need to identify which variables are set to zero in that solution. Basic variables are those that are not zero, while nonbasic variables are those that are zero. We will solve the system of equations to find the values of \( x_1 \), \( x_2 \), \( s_1 \), \( s_2 \), and \( s_3 \) for solution (J) and identify the nonbasic variables.

We are given a system of linear equations and a partially completed table of basic solutions. We need to determine which variables are nonbasic in basic solution (J).

The system of equations is: \[ \begin{array}{l} \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{s}_{1} = 28 \\ 2 x_{1}+x_{2}+s_{2} = 30 \\ 4 \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{s}_{3} = 40 \end{array} \]

In a basic solution, the number of basic variables is equal to the number of equations. Here, we have 3 equations, so there will be 3 basic variables and 2 nonbasic variables.

From the table, we see that in solution (J), \(s_2 = 0\) and \(s_3 = 0\). This implies that \(s_2\) and \(s_3\) are nonbasic variables. The remaining variables (\(x_1\), \(x_2\), and \(s_1\)) must be basic variables.

Final Answer