Henry Ernest Dudeney/Modern Puzzles/22 - Mrs. Wilson's Family/Working

Working for by : $22$ -- Mrs. Wilson's Family
The simultaneous equations in matrix form:


 * $\begin {pmatrix}

1 & 0 & -1 & 1 & -1 &  0 \\ 1 & 1 &  1 &  1 &  1 & -2 \\ 1 & 1 &  0 &  0 &  0 & -1 \\ 1 & 0 &  0 & -1 & -1 &  0 \\ 1 & 1 &  1 &  0 &  1 & -1 \\ 2 & 2 &  2 &  0 &  0 & -1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 30 \\ 75 \\ \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} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 21 \\ 18 \\ 18 \\ 9 \\ 12 \\ 39 \\ \end {pmatrix}$