\left\{ \begin{array} { l } { A ^ { 2 } + B ^ { 2 } = L B } \\ { 2 A + B = 3 X } \end{array} \right.
Solve for A, B, L, X (complex solution)
A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{C}\text{, }X\in \mathrm{C}
A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{C}\text{, }X\in \mathrm{C}
Solve for A, B, L, X
\left\{\begin{matrix}A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \begin{bmatrix}\frac{\sqrt{5}L+L}{6},\frac{-\sqrt{5}L+L}{6}\end{bmatrix}\text{; }A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \begin{bmatrix}\frac{\sqrt{5}L+L}{6},\frac{-\sqrt{5}L+L}{6}\end{bmatrix}\text{, }&L\leq 0\\A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\geq \frac{\sqrt{5}L+L}{6}\text{; }A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\geq \frac{\sqrt{5}L+L}{6}\text{, }&L>0\\A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \mathrm{R}\text{; }A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \mathrm{R}\text{, }&L\neq 0\\A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \begin{bmatrix}\frac{-\sqrt{5}L+L}{6},\frac{\sqrt{5}L+L}{6}\end{bmatrix}\text{; }A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\in \begin{bmatrix}\frac{-\sqrt{5}L+L}{6},\frac{\sqrt{5}L+L}{6}\end{bmatrix}\text{, }&L\geq 0\\A=\frac{3X-B}{2}\text{, }B=\frac{2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\geq \frac{-\sqrt{5}L+L}{6}\text{; }A=\frac{3X-B}{2}\text{, }B=\frac{-2\sqrt{L^{2}+3LX-9X^{2}}+2L+3X}{5}\text{, }L\in \mathrm{R}\text{, }X\geq \frac{-\sqrt{5}L+L}{6}\text{, }&L<0\end{matrix}\right.
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