Making Matrix Totally Unimodular

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Is there a way I can rewrite the following matrices to make the matrix (A) to be totally unimodular and still maintain the relevance of the equations. Thanks.

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Let $y_1=2x_1-x_6-x_7$, $y_2=3x_2-x_6-x_7-x_8$, $y_3=3x_3-x_7-x_8-x_9$, $y_4=3x_4-x_8-x_9-x_{10}$ and $y_5=2x_5-x_9-x_{10}$, then $A$ becomes the identity matrix plus a bunch of zeroes, when it multiplies the new vector $y_1,\dots,y_{10}$.