Solve {l}{T-m*g=m_{1}*a}{m_{2}*g-T=m_{2}*a}

$$\left\{ \begin{array} { l } { T - m \cdot g = m _ { 1 } \cdot a } \\ { m _ { 2 } \cdot g - T = m _ { 2 } \cdot a } \end{array} \right.$$

Solve for T, m, g, m_1, a, m_2

$$T=am_{1}\text{, }m\in \mathrm{R}\text{, }g=0\text{, }m_{1}=-m_{2}\text{, }a\in \mathrm{R}\text{, }m_{2}\in \mathrm{R}$$

$$T=0\text{, }m\in \mathrm{R}\text{, }g=0\text{, }m_{1}\in \mathrm{R}\text{, }a=0\text{, }m_{2}\in \mathrm{R}$$

$$T=gm+am_{1}\text{, }m=\frac{-am_{1}+gm_{2}-am_{2}}{g}\text{, }g\neq 0\text{, }m_{1}\in \mathrm{R}\text{, }a\in \mathrm{R}\text{, }m_{2}\in \mathrm{R}$$