I know that a 3-dimensional line can be represented by a symmetric equation - for example:
$$\frac{x-x_p}{x_q-x_p}=\frac{y-y_p}{y_q-y_p}=\frac{z-z_p}{z_q-z_p}$$
Why is this not written as a single equation, such as:
$$2\frac{x-x_p}{x_q-x_p}-\frac{y-y_p}{y_q-y_p}-\frac{z-z_p}{z_q-z_p} = 0$$
Any equation written as you have in the second equation defines a plane. What you have can be reduced to: $$c_1*x+c_2*y+c_3*z=c_4$$ where the $c$s are constants. In fact, you have three equations in the first equation and one in the second equation.
An example may help: take $x_p = y_p = z_p = 0,$ but $x_q = y_q = z_q = 1.$ The line in question is $$ x = y = z. $$ This is a line that goes through the origin $(0,0,0),$ and the point $(1,1,1).$
Your revision becomes $$ 2x - y - z = 0. $$ This plane passes though the origin $(0,0,0),$ and the point $(1,1,1),$ but it also passes through $(1,0,2).$ The line does not do that.
