Henry Ernest Dudeney/Puzzles and Curious Problems/158 - Newsboys/Working

Working for Puzzles and Curious Problems by Henry Ernest Dudeney: $158$ -- Newsboys

The simultaneous equations in matrix form:

$\begin {pmatrix}

-1 & 0 & 0 & 0 & 0 & 4 \\ -1 & 0 & 0 & 0 & 4 & 1 \\ -1 & 0 & 0 & 4 & 1 & 1 \\ -1 & 0 & 4 & 1 & 1 & 1 \\ -1 & 1 & 1 & 1 & 1 & 1 \\

0 &  0 & -1 &  1 & -1 &  1 \\

\end {pmatrix} \begin {pmatrix} n \\ J_J \\ J_C \\ S_N \\ J_B \\ S_T \end {pmatrix} = \begin {pmatrix} 4 \\ 4 \\ 4 \\ 4 \\ 0 \\ 100 \end {pmatrix}$

when converted to reduced echelon form, gives:

$\begin {pmatrix}

1 &  0 &  0 &  0 &  0 &  0 \\
0 &  1 &  0 &  0 &  0 &  0 \\
0 &  0 &  1 &  0 &  0 &  0 \\
0 &  0 &  0 &  1 &  0 &  0 \\
0 &  0 &  0 &  0 &  1 &  0 \\
0 &  0 &  0 &  0 &  0 &  1 \\

\end {pmatrix} \begin {pmatrix} n \\ J_J \\ J_C \\ S_N \\ J_B \\ S_T \end {pmatrix} = \begin {pmatrix} 1020 \\ 320 \\ 108 \\ 144 \\ 192 \\ 256 \end {pmatrix}$

Proof

\(\ds \)

\(\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} -1 & 0 & 0 & 0 & 0 & 4 & 4 \\ -1 & 0 & 0 & 0 & 4 & 1 & 4 \\ -1 & 0 & 0 & 4 & 1 & 1 & 4 \\ -1 & 0 & 4 & 1 & 1 & 1 & 4 \\ -1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & -1 & 1 & -1 & 1 & 100 \\ \end {array} }\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ -1 & 0 & 0 & 0 & 4 & 1 & 4 \\ -1 & 0 & 0 & 4 & 1 & 1 & 4 \\ -1 & 0 & 4 & 1 & 1 & 1 & 4 \\ -1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & -1 & 1 & -1 & 1 & 100 \\ \end {array} }\)

$r_1 \to -r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ 0 & 0 & 0 & 4 & 1 & -3 & 0 \\ 0 & 0 & 4 & 1 & 1 & -3 & 0 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & -1 & 1 & -1 & 1 & 100 \\ \end {array} }\)

$r_2 \to r_2 + r_1$, $r_3 \to r_3 + r_1$, $r_4 \to r_4 + r_1$, $r_5 \to r_5 + r_1$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ 0 & 0 & 0 & 4 & 1 & -3 & 0 \\ 0 & 0 & 4 & 1 & 1 & -3 & 0 \\ \end {array} }\)

rearranging the rows and $r_3 \to -r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ 0 & 0 & 0 & 4 & 1 & -3 & 0 \\ 0 & 0 & 0 & 5 & -3 & 1 & 400 \\ \end {array} }\)

$r_6 \to r_6 - 4 r_3$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 4 & 1 & -3 & 0 \\ 0 & 0 & 0 & 5 & -3 & 1 & 400 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ \end {array} }\)

rearranging the rows

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 5 & -3 & 1 & 400 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ \end {array} }\)

$r_4 \to r_4 / 4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & -\tfrac {17} 4 & \tfrac {19} 4 & 400 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ \end {array} }\)

$r_5 \to r_5 - 5 r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 4 & -3 & 0 \\ 0 & 0 & 0 & 0 & -\tfrac {17} 4 & \tfrac {19} 4 & 400 \\ \end {array} }\)

$r_5 \leftrightarrow r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 1 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & -\tfrac {17} 4 & \tfrac {19} 4 & 400 \\ \end {array} }\)

$r_5 \to r_5 / 4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 1 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & \tfrac {25} {16} & 400 \\ \end {array} }\)

$r_6 \to r_6 + \tfrac {17} 4 r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & -4 & -4 \\ 0 & 1 & 1 & 1 & 1 & -3 & -4 \\ 0 & 0 & 1 & -1 & 1 & -1 & -100 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 1 & -\tfrac 3 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 256 \\ \end {array} }\)

$r_6 \to \tfrac {16} {25} r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 1020 \\ 0 & 1 & 1 & 1 & 1 & 0 & 768 \\ 0 & 0 & 1 & -1 & 1 & 0 & 156 \\ 0 & 0 & 0 & 1 & \tfrac 1 4 & 0 & 192 \\ 0 & 0 & 0 & 0 & 1 & 0 & 192 \\ 0 & 0 & 0 & 0 & 0 & 1 & 256 \\ \end {array} }\)

$r_5 \to r_5 + \dfrac 3 4 r_6$, $r_4 \to r_4 + \dfrac 3 4 r_6$, $r_3 \to r_3 + r_6$, $r_2 \to r_2 + 3 r_6$, $r_1 \to r_1 + 4 r_6$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 1020 \\ 0 & 1 & 1 & 1 & 0 & 0 & 572 \\ 0 & 0 & 1 & -1 & 0 & 0 & -36 \\ 0 & 0 & 0 & 1 & 0 & 0 & 144 \\ 0 & 0 & 0 & 0 & 1 & 0 & 192 \\ 0 & 0 & 0 & 0 & 0 & 1 & 256 \\ \end {array} }\)

$r_4 \to r_4 - \dfrac 1 4 r_5$, $r_3 \to r_3 - r_5$, $r_2 \to r_2 - r_5$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 1020 \\ 0 & 1 & 1 & 0 & 0 & 0 & 428 \\ 0 & 0 & 1 & 0 & 0 & 0 & 108 \\ 0 & 0 & 0 & 1 & 0 & 0 & 144 \\ 0 & 0 & 0 & 0 & 1 & 0 & 192 \\ 0 & 0 & 0 & 0 & 0 & 1 & 256 \\ \end {array} }\)

$r_3 \to r_3 + r_4$, $r_2 \to r_2 + r_4$

\(\ds \)

\(\leadsto\)

\(\ds \paren {\begin {array} {cccccc{{|}}c} 1 & 0 & 0 & 0 & 0 & 0 & 1020 \\ 0 & 1 & 0 & 0 & 0 & 0 & 320 \\ 0 & 0 & 1 & 0 & 0 & 0 & 108 \\ 0 & 0 & 0 & 1 & 0 & 0 & 144 \\ 0 & 0 & 0 & 0 & 1 & 0 & 192 \\ 0 & 0 & 0 & 0 & 0 & 1 & 256 \\ \end {array} }\)

$r_2 \to r_2 - r_3$

$\blacksquare$