# The Slope of a Line

Inclination of a Line:
The angle from the X-axis to any given non horizontal line $l$ is called the inclination of line $l$. Here $\alpha$ is the inclination where $0 < \alpha < {180^ \circ }$, measured in a counter-clockwise direction from the positive X-axis to the line $l$.

Slope of a Line:
If $\alpha$ is the inclination of a non-vertical straight line $l$, then its slope or gradient is defined as $\tan \alpha$. The slope of a straight line is usually denoted by $m$, so the formula to find the slope of a line is given by
$\boxed{m = \tan \alpha }$

If a straight line is parallel to the X-axis, then its slope is equal to zero, i.e. $m = \tan {0^ \circ } = 0$. If the straight line is parallel to the Y-axis, then its slope is undefined, i.e. $m = \tan {90^ \circ } = \infty$.

If the line is parallel to the X-axis, then the ordinate of each point on the line is a fixed number, so its equation will be $y = a$, where $a$ is a fixed number. If the line is parallel to the Y-axis, then the abscissa of each point on the line is a fixed number, so its equation will be $x = a$, where $a$ is a fixed number.

Let ${m_1},{m_2}$ be the slopes of the lines ${l_1},{l_2}$ respectively.

(i) The lines ${l_1}$ and ${l_2}$ are parallel if and only if ${m_1} = {m_2}$

(ii) The lines ${l_1}$ and ${l_2}$ are perpendicular if and only if ${m_1} \times {m_2} = – 1$

Example: Find the slope of a straight line with inclination ${60^ \circ }$ with the X-axis.
Here we have the inclination $\alpha = {60^ \circ }$ with the X-axis, now we shall find the slope of the straight line using the formula $m = \tan \alpha$,

$m = \tan \alpha = \tan {60^ \circ } = \sqrt 3$