\left\{ \begin{array} { l } { M g + M a - T = M a _ { r } } \\ { T - m g - m a = m a _ { r } } \end{array} \right.
Solve for M, g, a, T, a_r, m
M=\frac{T}{a-a_{r}+g}\text{, }g=-a-a_{r}+\frac{T}{m}\text{, }a\in \mathrm{R}\text{, }T\neq 2a_{r}m\text{, }a_{r}\in \mathrm{R}\text{, }m\neq 0
M=0\text{, }g\neq a_{r}-a\text{, }a\in \mathrm{R}\text{, }T=0\text{, }a_{r}\in \mathrm{R}\text{, }m=0
M\in \mathrm{R}\text{, }g=-a\text{, }a\in \mathrm{R}\text{, }T=0\text{, }a_{r}=0\text{, }m\in \mathrm{R}
M\in \mathrm{R}\text{, }g=a_{r}-a\text{, }a\in \mathrm{R}\text{, }T=0\text{, }a_{r}\in \mathrm{R}\text{, }m=0
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