Henry Ernest Dudeney/Modern Puzzles/36 - More Bicycling/Working

Working for by : $36$ -- More Bicycling
The simultaneous equations in matrix form:


 * $\begin {pmatrix}

1 & 0 & -10 & 0 &   0 &  0 &  0 &   0 &  0 \\ 1 & 0 &   0 & -5 &   0 &  0 &  0 &   0 &  0 \\ 1 & 0 &   0 &  0 &  -3 &  0 &  0 &   0 &  0 \\ 0 & 1 &   0 &  0 &   0 & -4 &  0 &   0 &  0 \\ 0 & 1 &   0 &  0 &   0 &  0 & -8 &   0 &  0 \\ 0 & 1 &   0 &  0 &   0 &  0 &  0 &  -3 &  0 \\ 1 & 1 &  -4 &  0 &   0 & -4 &  0 &   0 &  4 \\ 1 & 1 &   0 & -5 &   0 &  0 & -5 &   0 &  5 \\ 1 & 1 &   0 &  0 & -12 &  0 &  0 & -12 & 12 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ t_{a_1} \\ t_{b_1} \\ t_{c_1} \\ t_{a_2} \\ t_{b_2} \\ t_{c_2} \\ t \\ \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 20 \\ 20 \\ 20 \\ \end {pmatrix}$

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

1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end {pmatrix} \begin {pmatrix} d_1 \\ d_2 \\ t_{a_1} \\ t_{b_1} \\ t_{c_1} \\ t_{a_2} \\ t_{b_2} \\ t_{c_2} \\ t \\ \end {pmatrix} = \begin {pmatrix} 200/27 \\ 40/27 \\ 20/27 \\ 40/27 \\ 200/81 \\ 10/27 \\ 5/27 \\ 40/81 \\ 35/9 \\ \end {pmatrix}$