Question

How many intercepts can a line have?

Answer 1

Hint: 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.