Henry Ernest Dudeney/Puzzles and Curious Problems/134 - The Bag of Nuts/Working

== Working for by : $134$ -- The Bag of Nuts ==

The simultaneous equations in matrix form:


 * $\begin {pmatrix}

1 & 1 &  1 &  1 &  1 \\ 1 &  1 &  0 &  0 &  0 \\ 0 &  1 &  1 &  0 &  0 \\ 0 &  0 &  1 &  1 &  0 \\ 0 &  0 &  0 &  1 &  1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \end {pmatrix} = \begin {pmatrix} 100 \\ 52 \\ 43 \\ 34 \\ 30 \end {pmatrix}$

when converted to reduced echelon form, gives:
 * $\begin {pmatrix}

1 & 0 &  0 &  0 &  0 \\ 0 &  1 &  0 &  0 &  0 \\ 0 &  0 &  1 &  0 &  0 \\ 0 &  0 &  0 &  1 &  0 \\ 0 &  0 &  0 &  0 &  1 \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \end {pmatrix} = \begin {pmatrix} 27 \\ 25 \\ 18 \\ 16 \\ 14 \end {pmatrix}$