The result of your simplification is a single equation, which defines a plane in 3D space (two variables are free and one depends on them). The original formula consists of two first-order equations. Each of them determines a plane, and together they determine the intersection of those planes, i.e., a line.

Suppose the following equations are given.

$$

\frac{x-x_0}{A}=\frac{y-y_0}{B} =\frac{z-z_0}{C}

$$

If we denote the value of these three fractions by $t$ and express $x,y,z$ through $t$, we get a system of parametric equations

$$

\begin{aligned}

x&=x_0+At\\

y&=y_0+Bt\\

z&=z_0+Ct

\end{aligned}

$$

They determine a line passing through $(x_0,y_0,z_0)$ in the direction $(A,B,C)$.