The problem from Artin's book states:
Let $$A=\begin{bmatrix} a&b\\ c&d\\ \end{bmatrix}$$ be a real $2\times 2$ matrix. The condition that a column vector $V$ be an eigenvector for left multiplication by $A$ is that $AX = Y$, which means that the slopes $s= {x_2\over x_1} $ and $s' = {y_2 \over y_1}$ are equal.
The question is to find the equation in $s$ that expresses this equality.
My attempt:
$$AX=Y=\lambda X$$ Thus, $$A=\begin{bmatrix} a&b\\ c&d\\ \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix} = \begin{bmatrix} ax_1+bx_2\\ cx_1+dx_2\\ \end{bmatrix} $$
Therefore, $$ax_1+bx_2=y_1=\lambda x_1$$ $$cx_1+dx_2=y_2=\lambda x_2$$
The next step is to divide each equation by $x_1$ and get: $$ a + b{x_2 \over x_1} = {y_1\over x_1} = \lambda $$ $$ c + d{x_2 \over x_1} = {y_2\over x_1} = \lambda {x_2 \over x_1} $$
Then I am dividing $2^{nd}$ equation by the $1^{st}$ one and getting: $${c \over a} + {d \over b} = {y_2 \over y_1} = {x_2 \over x_1} $$
Any corrections and further suggestions are more than welcome. Thank you in advance for your effort.