How to Find the x-Intercept of a Line?

The \(x\)-intercept for any curve is the value of the \(x\) coordinate of the point where the graph cuts the \(x\)-axis. The following guide learn you how to find the \(x\)-intercept of a line.

How to Find the x-Intercept of a Line?

The \(x\)-intercept is the value of the \(x\) coordinate of a point where the value of the \(y\) coordinate is equal to zero.

Step by step guide to finding the \(x\)-intercept

In the linear equation, we read that the general form is represented by \(y=mx + b\) that \(m\) and \(b\) are constant. The \(x\)-intercept is the point or the coordinate from which the line passes and that lies at the \(x\)-axis of the plane. This means the \(y\) coordinate value of the respective linear equation will always be equal to \(0\) when it passes through the \(x\)-axis. The \(x\)-intercept is also known as the horizontal intercept.

\(x\)-intercept formula

There are various formulas and equations for intercepts. Below are some of the most commonly used formulas. All these formulas are obtained by substituting \(y = 0\) in the equation and solving for \(x\).

  • The general form of a straight line is \(ax+by+c=0\),  where \(a,b,c\) are constants. The \(x\)-intercept of the line can be obtained by putting \(y = 0\), \(x\)-intercept \(=-\frac{c}{a}\).
  • The slope-intercept form of a straight line is as follows: \(y = mx+c\), where \(m\) is the slope of the line and \(c\) is the \(y\)-intercept. The \(x\)-intercept of the line can be obtained by putting \(y=0\), \(x\)-intercept \(= −\frac{c}{m}\).
  • The point-slope form of a straight line is \(y−b = m (x−a)\), where \(m\) is the slope of the line, \((a,b)\) is a point on the line. The \(x\)-intercept of the line can be obtained by putting \(y = 0\), \(x\)-intercept \(= \frac{(am−b)}{m}\).
  • The intercept form of a straight line is \(\frac{x}{a} +\frac{y}{b} = 1\) where \((a, 0)\) is its \(x\)-intercept and \((0, b)\) is its \(y\)-intercept.

\(x\)-intercept on a graph

To find the \(x\)-intercept of a line of the form \(y = mx+b\), substitute \(y = 0\). For example to calculate the \(x\)-intercept of the line \(y = 2x−4\), put \(y = 0\) in the equation of a line:

\(0=2x-4\)

\(4=2x\)

\(x=2\)

So, the \(x\)-intercept of the line \(y=2x−4\) is \(2\).

Find the equation of a line using \(x\)-intercept

Consider an \(x\)-intercept example to understand the process of finding the \(x\)-intercept.

Example: Find the equation of a line with the slope equal to \(2\) and the \(x\)-intercept equal to \(-5\).

Solution: The general equation of a line with slope \(m\) is \(y=mx+c\). The \(x\)-intercept of the line is \(−5\). Using the formula of the \(x\)-intercept:

\(x=-\frac{c}{m}\)

\(-5=-\frac{c}{2}\)

\(c=10\)

By putting the value of \(c\) and \(m\) we have, \(y=2x+10\).

Finding the \(x\)-Intercept – Example 1:

Find the \(x\)-intercept of the equation \(y=-3x – 4\).

Solution:

To find the \(x\)-intercept, set \(y=0\):

\(y=-3x-4\)

\(0=-3x-4\)

\(3x=-4\)

\(x=-\frac{4}{3}\)

Exercises for Finding the \(x\)-Intercept

  1. A line with slope \(-2\) passes through point \(P(a,1)\). If the sum of both the intercepts of the line is equal to \(6\), find the value of \(a\).
  2. Find the \(x\)-intercept of \(y=x^2-3x+2\).
  3. Find the \(x\)-intercept of \(y=x^2-4\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{\frac{3}{2}}\)
  2. \(\color{blue}{(2,0),(1,0)}\)
  3. \(\color{blue}{(2,0),(-2,0)}\)

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