CAT 2025 Lesson : Trigonometry & Coordinate - Equation of a Line

3.3 Slope of a line

The slope of a line determines the flatness or steepness of the line with respect to the axes. Positive and negative slopes indicate the direction that the line takes with respect to the $y$-axis.

Where the angle formed by the line with the $x$-axis is θ, Slope ($m$) = tan θ

Where ($x_1, y_1$) and ($x_2, y_2$) are two points that the line passes through (and $x_1 ≠ x_{2}$), Slope ($m$) = $\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}$

The slope gives us the rate of change of $y$ with respect to the change of $x$. For instance, if the slope of a line is 3, then 1 unit increase in the $x$-coordinate will result in a 3 units increase in the $y$-coordinate along the line. Similarly, if the slope is -0.5, then 1 unit increase in $x$-coordinate will result in 0.5 decrease in the $y$-coordinate (decrease in this case as the slope is negative).

3.4 Lines parallel to x-axis & y-axis

As shown below, few of the points that a line that is parallel to $x$-axis passes through are $(-2, k), (-1, k), (0, k), (1, k), (2, k)$. The $y$-coordinate of the points never changes. Therefore, the equation of such a line is $y = k$.



A vertical line parallel to the $y$-axis, as shown below, passes through are $(k, -2), (k, -1), (k, 0), (k, 1), (k, 2)$. The $x$-coordinate of the points never changes. Therefore, the equation of such a line is $x = k$.

3.5 Equation of a Line

Equations of a line can be formed or written in the following ways:

1) Slope & y-intercept form: $y = mx + c$
This is one of the easiest ways to form an equation. $m$ is the slope of the line and $c$ is the $y$-intercept (note that when $x = 0, y = c$ and therefore, $c$ is the $y$-intercept).

$m$ indicates the direction and steepness of the line while $c$ indicates the point where the line cuts the $y$-axis. An equation represented this way would help us to quickly draw or visualise the line.

2) $\dfrac{x}{a} + \dfrac{y}{b} = 1$

where $a$ and $b$ are the $x$-intercept and $y$-intercept. This can be used when the two intercepts are known.

3) $ax + by + c = 0$

This is a general form in which an equation of the line is represented.
(a) If $x = 0, y = - \dfrac{c}{b}$ which is the $y$-intercept

(b) If $y = 0, x = - \dfrac{c}{a}$ which is the $x$-intercept

(c) Equation can be re-written as $y = - \dfrac{ax}{b} - \dfrac{c}{b}$ where the Slope is $\dfrac{-a}{b}$

(a), (b) and (c) need not be memorised as you should be able to quickly deduce these.