Question

How many intercepts can a line have?

Answer 1

Here, will use the slope intercept form of a line to find the intercepts in a line. A line is a one dimensional figure which has the length but not the width. A line can be in any direction but it should be straight without any deviation at any point. A line can be formed only by a straight set of points.

Formula Used:
A straight line can be represented in the slope Intercept form as$y = mx + c$ where$m$ is the slope and $c$ is the $y$-intercept respectively.

Complete Step by Step Solution:
We know that a line can extend its length in both the directions infinitely.
A straight line can be represented in the slope Intercept form as $y = mx + c$ where $m$ is the slope and $c$ is the $y$-intercept respectively.
Now, by substituting $x = 0$ in the slope intercept form, we get
$\Rightarrow y = m\left( 0 \right) + c$
$\Rightarrow y = c$
Now, by substituting $y = 0$ in the slope intercept form, we get
$\Rightarrow 0 = mx + c$
$\Rightarrow x = - \dfrac{c}{m}$
Thus $x$ intercept and $y$ intercept are possible only if $m \ne 0$ in the slope intercept form.
Now, by substituting $c = 0$ in the slope intercept form, we get the $x$intercept $x = 0$and the $y$ intercept $y = 0$
If $c = 0$, then the $x$ intercept and the $y$ intercept is at the origin $\left( {0,0} \right)$.
If the line $x = 0$, then the line has an infinite number of intercepts with the $y$ axis and has only one intercept with the $x$ axis.
If the line $y = 0$, then the line has an infinite number of intercepts with the$x$axis and has only one intercept with the $y$ axis.

Therefore, a line can have an infinite number of intercepts with the $x$ axis or $y$ axis.

Note:
We know the intercepts are defined as a graph which crosses either the $x$ axis or the $y$ axis. Also all the graphs of a function will have the intercepts, but the graph of the linear function will have both the intercepts. We know that a point crossing the $x$-axis, it is called $x$-intercept and a point crossing the y-axis is called the y-intercept. We should note that a line will definitely possess at least an intercept with the $x$ axis and with the $y$ axis.