Lines Parallel to the Axes

Before building the general forms of a line's equation, it is worth isolating two special — but very common — cases directly from the definition of a straight line as the set of points satisfying one fixed condition.

Line parallel to the x-axis. A line parallel to (or coincident with) the x-axis consists of every point whose y-coordinate is the same fixed number, say $b$, regardless of $x$ — this is exactly what "parallel to a horizontal axis" means: constant height. So its equation is

Every such line has slope $m=0$ (inclination $\theta=0^\circ$), consistent with Section 2. The x-axis itself is the special case $b=0$, i.e. $y=0$.

Line parallel to the y-axis. By the same reasoning, a line parallel to (or coincident with) the y-axis consists of every point whose x-coordinate is the same fixed number, say $a$, regardless of $y$. Its equation is

Such a line has an undefined slope (inclination $\theta=90^\circ$), since any two points on it give $x_2-x_1=0$ in the slope formula, making the denominator zero. The y-axis itself is the special case $a=0$, i.e. $x=0$.

Reading off $a$ and $b$. In both cases, the constant is read directly off the one point through which the line is required to pass: a line through $(x_1,y_1)$ parallel to the x-axis is simply $y=y_1$, and a line through the same point parallel to the y-axis is simply $x=x_1$ — no computation needed beyond writing down the coordinate that stays fixed.

Worked illustration. The line through $(5,-2)$ parallel to the x-axis is $y=-2$; the line through the same point parallel to the y-axis is $x=5$. These two lines are themselves perpendicular to each other (one horizontal, one vertical), meeting exactly at $(5,-2)$. …