Solve -lambda=x/a=y/b=z/c

$$- \lambda = \frac { x } { a } = \frac { y } { b } = \frac { z } { c }$$

Evaluate (complex solution)

$$-\lambda =\frac{x}{a}\text{ and }\frac{x}{a}=\frac{y}{b}\text{ and }\frac{y}{b}=\frac{z}{c}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{az}{c}\text{, }&a\neq 0\text{ and }\lambda =-\frac{z}{c}\text{ and }c\neq 0\text{ and }b=\frac{cy}{z}\text{ and }z\neq 0\text{ and }y\neq 0\\x=0\text{, }&a\neq 0\text{ and }\lambda =0\text{ and }b\neq 0\text{ and }z=0\text{ and }y=0\text{ and }c\neq 0\end{matrix}\right.$$

Solve for λ

$$\left\{\begin{matrix}\lambda =0\text{, }&a\neq 0\text{ and }y=0\text{ and }x=0\text{ and }b\neq 0\text{ and }z=0\text{ and }c\neq 0\\\lambda =-\frac{z}{c}\text{, }&a=\frac{cx}{z}\text{ and }x\neq 0\text{ and }z\neq 0\text{ and }y\neq 0\text{ and }b=\frac{cy}{z}\text{ and }c\neq 0\end{matrix}\right.$$