Write the equation of the line through M_1(x_1,y_1,z_1) and…
Write the equation of the line through $M_1(x_1,y_1,z_1)$ and $M_2(x_2,y_2,z_2)$.
Explanation of Method
The equations of the line through two given points in space.
Step 1: Direction
\[ \vec p=\overrightarrow{M_1M_2}=\{x_2-x_1,\ y_2-y_1,\ z_2-z_1\}\qquad(M_1\ne M_2) \]
Step 2: Equations
\[\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\]
Parametric form: \[x=x_1+t(x_2-x_1),\quad y=y_1+t(y_2-y_1),\quad z=z_1+t(z_2-z_1)\]
Here $t=0$ gives $M_1$, $t=1$ gives $M_2$, and $0\le t\le1$ is the segment $M_1M_2$.
Step 3: Zero denominators
A zero denominator means that coordinate is constant along the line. For example, if $x_2=x_1$: $x=x_1,\qquad \frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$.
If two denominators vanish, e.g. $x_2=x_1$ and $y_2=y_1$, the line is $x=x_1,\ y=y_1$, parallel to $Oz$.
Result
\[\boxed{\,\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\,}\]
Remark
Example: through $(1,0,2)$ and $(3,-1,2)$ we get $\tfrac{x-1}{2}=\tfrac{y}{-1}$, $z=2$. This line lies in the plane $z=2$.