<s>[INST] Linear programming exercise I want some help for the following exercise



I know that the leaving variable is the basic variable associated with the smallest nonnegative ratio with the strictly positive denominator. I can't understand how the current basic variables ($x_5$, $x_6$, $x_7$, $x_8$) are matched with the equations.
 [/INST] Let $x_r$ and $x_j$ denote the leaving and entering variables respectively.  ($r$ stands for "row"; $j$ runs through columns.)
Write the simplex tableaux.
$$
\require{enclose}
% inner horizontal array of arrays
\begin{array}{cc}
% inner array of minimum values
\begin{array}{r|r|r|r}
\text{basis} & a_j & b & b/a_j \\
\hline
5 & 1 & 4 & 4 \\
6 & 5 & 8 & 8/5 \\
\to7 & 2 & 3 & \enclose{circle}{3/2} \\
8 & * & 0 & -
\end{array}
&
% inner array of minimum values
\begin{array}{r|r|r|r}
\text{basis} & a_j & b & b/a_j \\
\hline
5 & 2 & 4 & 2 \\
6 & * & 8 & - \\
\to7 & 3 & 3 & \enclose{circle}{1} \\
8 & * & 0 & -
\end{array} \\
j=1,r=7 & j=2,r=7 \\
% inner array of minimum values
\begin{array}{r|r|r|r}
\text{basis} & a_j & b & b/a_j \\
\hline
5 & * & 4 & - \\
6 & * & 8 & - \\
7 & * & 3 & - \\
\to8 & 1 & 0 & \enclose{circle}{0}
\end{array}
&
% inner array of minimum values
\begin{array}{r|r|r|r}
\text{basis} & a_j & b & b/a_j \\
\hline
\to5 & 5 & 4 & \enclose{circle}{4/5} \\
6 & 6 & 8 & 4/3 \\
7 & 3 & 3 & 1 \\
8 & * & 0 & -
\end{array} \\
j=3,r=8 & j=4,r=5
\end{array}
$$
Fix $j$.  Observe that $r$ is the index which minimises $\{b/a_{rj} \mid a_{rj}>0 \}$.  (You may refer to a theoretical explanation for such choice.)
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