Rewrite a system of linear equations
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Fix $r$ and $dequiv r+binom{r}{2}$ and consider $d$ real numbers $b_1,b_2,..., b_d$ . I have the following system of linear equations: $$ begin{cases} text{$(diamond)$ Differences between the elements ${b_1,b_h}$ for $h=2,...,r$ $hspace{1cm}$ ($r-1$ elements)}\ b_{r+1}=b_1-b_2\ b_{r+2}=b_1-b_3\ ...\ b_{2r-1}=b_1-b_r\ -------\ text{$(diamond)$ Differences between the elements ${b_2,b_h}$ for $h=3,...,r$ $hspace{1cm}$ ($r-2$ elements)}\ b_{2r}=b_2-b_3\ ...\ b_{3r-3}=b_2-b_r\ -------\ text{$(diamond)$ Etc. }\ ....\ -------\ text{$(diamond)$ Differences between the elements ${b_{r-1},b_h}$ for $h=r$ $hspace{1cm}$ ($1$ element)} \ b_d=b_{r-1}-b_r end{cases} $$ Question: For any $r$ , I want to re-write in an EQUIVALENT way such that it has only relations with the $+$ s
