Henry Ernest Dudeney/Modern Puzzles/13 - Find the Coins/Working
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Working for Modern Puzzles by Henry Ernest Dudeney: $13$ -- Find the Coins
The simultaneous equations in matrix form:
- $\begin {bmatrix} 4 & -4 & -4 \\ -2 & 6 & -2 \\ -1 & -1 & 7 \end {bmatrix} \begin {bmatrix} a \\ b \\ c \end {bmatrix} = \begin {bmatrix} 120 \\ 120 \\ 120 \end {bmatrix}$
when converted to reduced echelon form, gives:
- $\begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {bmatrix} \begin {bmatrix} a \\ b \\ c \end {bmatrix} = \begin {bmatrix} 195 \\ 105 \\ 60 \end {bmatrix}$
Proof
\(\ds \) | \(\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 4 & -4 & -4 & 120 \\ -2 & 6 & -2 & 120 \\ -1 & -1 & 7 & 120 \end {array} }\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & -1 & 30 \\ -1 & 3 & -1 & 60 \\ -1 & -1 & 7 & 120 \end {array} }\) | $r_1 \to r_1 / 4$, $r_2 \to r_2 / 2$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & -1 & 30 \\ 0 & 2 & -2 & 90 \\ 0 & -2 & 6 & 150 \end {array} }\) | $r_2 \to r_2 + r_1$, $r_3 \to r_3 + r_1$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & -1 & 30 \\ 0 & 1 & -1 & 45 \\ 0 & -1 & 3 & 75 \end {array} }\) | $r_2 \to r_2 / 2$, $r_3 \to r_3 / 2$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & -1 & 30 \\ 0 & 1 & -1 & 45 \\ 0 & 0 & 2 & 120 \end {array} }\) | $r_3 \to r_3 + r_2$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & -1 & 30 \\ 0 & 1 & -1 & 45 \\ 0 & 0 & 1 & 60 \end {array} }\) | $r_3 \to r_3 / 2$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & -1 & 0 & 90 \\ 0 & 1 & 0 & 105 \\ 0 & 0 & 1 & 60 \end {array} }\) | $r_1 \to r_1 + r_3$, $r_2 \to r_2 + r_3$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {\begin {array} {ccc{{|}}c} 1 & 0 & 0 & 195 \\ 0 & 1 & 0 & 105 \\ 0 & 0 & 1 & 60 \end {array} }\) | $r_1 \to r_1 + r_2$ |
$\blacksquare$