A Simple Derivation for the equation of a linear equation in two variables. In school, I have been learning that the standard expression of a linear equation in two variables is of the form: $ax + by + c = 0$ while $a \neq 0$ and $b \neq 0$. I want to understand the purpose of the constant 'c' in this equation and where it was derived from, whether it be graphically or algebraically.
 A: Let's start with the fact that we want linear equations in two variables to represent a line. 
Suppose we didn't have a constant $c$. If we considered all equations of the form $ax+by=0$, we would only get lines through the origin, as follows: we could solve our equation to get $y=-\frac{a}{b}x$, which always passes through $(0,0)$. 
Adding the constant $c$ allows us to consider all lines in the plane. Specifically the equation $ax+by+c=0$ passes through the points $(0, -\frac{c}{b})$ and $(-\frac{c}{a},0)$
A: $ax+by=0$ is an equation of a line passing through the origin. This can be verified by putting $x=y=0$ in the equation. LHS=RHS and hence verified. If however, you want an equation of a line which doesn't pass through the origin, the factor 'c' is needed. 
In the more 'popular' form of the equation $y = mx+c$ (Just a fancy way of arranging $ax+by+c=0$ to include slope and intercept), c becomes the intercept of the line on the Y axis. In other words, when $x=0$, $y = c$. The factor c only adds to "move" the line parallel to itself up or down. See below. There are 2 lines parallel to each other (Lets call them upper and lower). One of them is $y = 2x + 3 (Upper)$ and the other $y = 2x (Lower)$ The factor 3 "moves" the line up by 3 units when the line intersects the Y axis.
 
A: Not quite sure what you're asking about fundamental principles.  Do you mean more or less from the definition of a line?  Well, if you define a line as having constant slope, you can write this as
$\frac{y-y_1}{x-x_1}=m$
which can then be manipulated to
$y-y_1=m(x-x_1),y=mx+y_1-mx_1$
If you plug in the point $(0,b)$ for $(x_1,y_1)$, you get the slope-intercept form
$y=mx+b$
From here, you could subtract both sides by $mx$ and multiply by a constant to clear fractions if you wish to get the form you desire.
