Equation of a hyperplane in two dimensions I'm having a bit of confusion with a passage I'm reading about Maximal Margin Classifiers in the context of Support Vector Machines, which is making me think I need to go back to grade school. Here's the passage:

For some reason I can't seem to understand why this is the equation of a line. If I use the basic equation I know, $y=mx+b$, then rename $x$ to $X_{1}$, rename $y$ to $X_{2}$, rename $b$ to $\beta_{0}$, rename $m$ to $\beta_{1}$, then I get $X_{2}=\beta_{1}X_{1}+\beta_{0}$, or rearranged as: $\beta_{0}+\beta_{1}X_{1}-X_{2}=0$. Now flipping the sign I could maybe ignore as being unimportant, but why is there an extra coefficient term as well?
 A: You could just multiply the equation $\beta_0+\beta_1X_1-X_2=0$ by another constant to give a "visible" coefficient on the $X_2$ as in equation (9.1).
More important, however, is that $y=mx+b$ is not the general equation of a line, because it will never give you a vertical line $x=c$.  The form (9.1) is superior in this respect: it will give you vertical lines (when $\beta_2=0$), horizontal lines (when $\beta_1=0$), and everything in between.
(9.1) has the minor disadvantage that different equations will represent the same line, for example,
$$1+2X_1+3X_2=0\quad\hbox{and}\quad 10+20X_1+30X_2=0\ ,$$
but as long as you are aware of this you will find that it rarely causes difficulties.
A: A two-dimensional plane $P$ is spanned by $2$-linearly independent vectors say $X_1,X_2$. Since $\mathbb{R}^2$ is two dimensional, then any point $(u,v)$ is in the image of the matrix $A = [X_1 \mid X_2]$ i.e,
$$ \begin{pmatrix} X_1 & X_2 \end{pmatrix} \begin{pmatrix} \beta_1 \\ \beta_2\end{pmatrix} = \begin{pmatrix} u \\ v\end{pmatrix}$$
It we call $(u,v)^T $  say $C_0$ then the above says,
$$ X_1 \beta_1 + X_2 \beta_2 - C_0 = 0$$
Now set $-C_0 = \beta_0$ then we have,
$$X_1 \beta_1 +X_2 \beta_2 + \beta_0 = 0$$
