Conditions for Homogeneity/Straight Line

Theorem
The line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ is homogeneous $\beta = 0$.

Sufficient Condition
Let the line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ be homogeneous.

Then the origin $\tuple {0, 0}$ lies on $L$.

Hence:

Necessary Condition
Let the equation of $L$ be $\alpha_1 x_1 + \alpha_2 x_2 = 0$.

Then:
 * $0 = \alpha_1 0 + \alpha_2 0 \in L$

and so:
 * $\tuple {0, 0}$ lies on $L$

Hence $L$ is homogeneous by definition.