Solve {l}{qU=1/2mv^2}{qVB=mv^2/r}

$$\left\{ \begin{array} { l } { q U = \frac { 1 } { 2 } m v ^ { 2 } } \\ { q V B = \frac { m v ^ { 2 } } { r } } \end{array} \right.$$

Solve for q, U, m, v, V, B, r (complex solution)

$$\left\{\begin{matrix}\\r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m\in \mathrm{C}\text{, }v=0\text{, }V\neq 0\text{, }B\neq 0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m=0\text{, }v\in \mathrm{C}\text{, }V\neq 0\text{, }B\neq 0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m\in \mathrm{C}\text{, }v=0\text{, }V\in \mathrm{C}\text{, }B=0\text{; }r\neq 0\text{, }q\in \mathrm{C}\text{, }U=0\text{, }m\in \mathrm{C}\text{, }v=0\text{, }V\in \mathrm{C}\text{, }B=0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m\in \mathrm{C}\text{, }v=0\text{, }V=0\text{, }B\in \mathrm{C}\text{; }r\neq 0\text{, }q\in \mathrm{C}\text{, }U=0\text{, }m\in \mathrm{C}\text{, }v=0\text{, }V=0\text{, }B\in \mathrm{C}\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m=0\text{, }v\in \mathrm{C}\text{, }V\in \mathrm{C}\text{, }B=0\text{; }r\neq 0\text{, }q\in \mathrm{C}\text{, }U=0\text{, }m=0\text{, }v\in \mathrm{C}\text{, }V\in \mathrm{C}\text{, }B=0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{C}\text{, }m=0\text{, }v\in \mathrm{C}\text{, }V=0\text{, }B\in \mathrm{C}\text{; }r\neq 0\text{, }q\in \mathrm{C}\text{, }U=0\text{, }m=0\text{, }v\in \mathrm{C}\text{, }V=0\text{, }B\in \mathrm{C}\text{, }&\text{unconditionally}\\r\neq 0\text{, }q=\frac{mv^{2}}{BVr}\text{, }U\neq 0\text{, }m\in \mathrm{C}\text{, }v\in \mathrm{C}\text{, }V=\frac{2U}{Br}\text{, }B\neq 0\text{, }&U\neq 0\text{ and }B\neq 0\text{ and }r\neq 0\end{matrix}\right.$$

Solve for q, U, m, v, V, B, r

$$\left\{\begin{matrix}\\r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m\in \mathrm{R}\text{, }v=0\text{, }V\neq 0\text{, }B\neq 0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m=0\text{, }v\neq 0\text{, }V\in \mathrm{R}\setminus \frac{2U}{Br}\text{, }r\neq 0\text{ and }B\neq 0\text{ and }U\neq 0,0\text{, }B\neq 0\text{; }r\neq 0\text{, }q=0\text{, }U=0\text{, }m=0\text{, }v\neq 0\text{, }V\neq 0\text{, }B\neq 0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m\in \mathrm{R}\text{, }v=0\text{, }V\in \mathrm{R}\text{, }B=0\text{; }r\neq 0\text{, }q\in \mathrm{R}\text{, }U=0\text{, }m\in \mathrm{R}\text{, }v=0\text{, }V\in \mathrm{R}\text{, }B=0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m\in \mathrm{R}\text{, }v=0\text{, }V=0\text{, }B\in \mathrm{R}\text{; }r\neq 0\text{, }q\in \mathrm{R}\text{, }U=0\text{, }m\in \mathrm{R}\text{, }v=0\text{, }V=0\text{, }B\in \mathrm{R}\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m=0\text{, }v\in \mathrm{R}\text{, }V\in \mathrm{R}\text{, }B=0\text{; }r\neq 0\text{, }q\in \mathrm{R}\text{, }U=0\text{, }m=0\text{, }v\in \mathrm{R}\text{, }V\in \mathrm{R}\text{, }B=0\text{; }r\neq 0\text{, }q=0\text{, }U\in \mathrm{R}\text{, }m=0\text{, }v\in \mathrm{R}\text{, }V=0\text{, }B\in \mathrm{R}\text{; }r\neq 0\text{, }q\in \mathrm{R}\text{, }U=0\text{, }m=0\text{, }v\in \mathrm{R}\text{, }V=0\text{, }B\in \mathrm{R}\text{, }&\text{unconditionally}\\r\neq 0\text{, }q=\frac{mv^{2}}{BVr}\text{, }U\neq 0\text{, }m\in \mathrm{R}\text{, }v\in \mathrm{R}\text{, }V=\frac{2U}{Br}\text{, }B\neq 0\text{, }&r\neq 0\text{ and }B\neq 0\text{ and }U\neq 0\end{matrix}\right.$$