X Intercept – Definition, Formula, Graph, Examples

What Is X–intercept?

The x-intercept is a point where the graph of a function or a curve intersects with the x-axis of the coordinate system.

So, what does x-intercept mean on the Cartesian plane? The value of the x-coordinate of a point where the value of y-coordinate is equal to zero is known as the x–intercept. The x-intercept is also called “horizontal intercept.”

X-Intercept Definition

For any line or a curve, the point at which the graph cuts the x-axis is called the x-intercept. For the x-coordinate, the value of the y-coordinate is zero.

X-Intercept on a Graph

Let’s understand how to find x intercept on a graph. X-intercept is a point where the line crosses the x-axis.

Consider the line shown in the graph below. The line cuts the x-axis at point (5,0). Thus, the x-intercept of the line is 5.

X-Intercept Formula

We can find the x-intercept by substituting $y = 0$ in the equation of line. Let’s see how to get the x-intercept in terms of different forms of the equation of a line.

• The general form of a straight line is given by $ax + by + c = 0$, where a,b,c are constants. 

If we substitute $y = 0$, we get

$ax + b.0 + c = 0$

$ax + c = 0$

$ax = \;-\; c$

$x = \frac{- c}{a}$

• The slope-intercept form of a straight line is given by $y = mx + c$, where m is the slope of the line and c is the y-intercept. 

If we substitute $y = 0$, we get

$0 = mx + c$

$-\;c = mx$

$x = \;-cm$

• The point-slope form of a straight line is given by $y\;-\;b = m(x\;-\;a)$, where m is the slope of the line and (a,b) is a point on the line.

If we substitute $y = 0$, we get

$0\;-\;b = m(x\;-\;a)$

$\;-\;b = mx\;-\;am$

$am\;-\;b = mx$

$x = \frac{am\;-\;b}{m}$

• The intercept form of a straight line is given by $\frac{x}{a} + \frac{y}{b} = 1$, where a is its x-intercept and b is its y-intercept.

How to Find the X-Intercept

Let’s discuss three methods to find the x-intercept.

Finding X-intercept Using the Graph

Let’s understand how to find x-intercept on a graph. Consider the graph of a line given below:

We can find the x-intercept from the graph by finding the point where the line touches the x-axis. 

In this case, the line cuts the x-axis at 7.

So, x-intercept $= 7$

Finding X-intercept Using the Equation of a Line

The equation of a line is given by$ax + by + c = 0$.

Suppose we have a equation of a line $3x + 4y = 12$.

On converting it into general form, we get $3x + 4y\;-\;12 = 0$

x-intercept $= \frac{-c}{a} = \frac{-(-12)}{3} = 4$

Finding X-intercept Using the Quadratic Formula

The quadratic formula is given by: $ax^2 + bx + c = 0$

We can find the x-intercept using the formula: $x = \frac{-b \pm \sqrt{b^2 \;-\; 4ac}}{2a}$

Example:  $x^2\;-\;5x + 6 = 0$

Here, $a = 1,b = \;-\;5, c = 6$

$x = \frac{5 \pm \sqrt{25 \;-\; 24}}{2} = \frac{5 \pm 1}{2}$

$x = \frac{4}{2},\;\frac{6}{2}$

$x = 2,\;3$

Finding Equation of a Line Using X-Intercept and Slope

Let’s understand how to find the equation of a line using x-intercept and slope with an example.

Example: Find the equation of a line with the slope = 3 and the x-intercept $= -4$.

We know that the general equation of a line with slope m is given by

 $y = mx + b$

Using the formula of the x-intercept

$x = \frac{-b}{m}$

$-4 = \frac{-b}{3}$

$b = 12$

On substituting the value of c and m, we have

$y = 3x + 12$

Facts about X-intercept

Solved Examples on X-intercept

1. Find the x-intercept of the line $5x\;-\;6y + 15 = 0$.

Solution: 

The equation of the line of the form $ax + by + c = 0$ has the x-intercept as $\frac{-c}{a}$.

In $5x\;-\;6y + 15 = 0,\; a = 5,\; b = -6,\; c = 15$

x – intercept $= \;-\;ca = -155 = \;-3$

Another way:

Simply substitute $y = 0$.

$5x\;-\;6(0) + 15 = 0$

$5x = -15$

$x = – 3$

2. What is the x-intercept of the quadratic equation given by: $2x^2 + 7x\;-\;9 = 0$?

Solution: 

The quadratic formula of the form ax2+bx+c=0 has the x-intercept as $x = \frac{\;-b\pm \sqrt{b^2 – 4ac}}{2a}$.

Here, $a = 2,\;b = 7,\;c = -9$

So, x – intercept $= \frac{- \pm \sqrt{749\;-\;4\times 2\times (-9)}}{2\times2} = \frac{-7\pm \sqrt{49 + 72}}{2\times2} = \frac{-7\pm\sqrt{121}}{4} = \frac{-7\pm11}{4}$

$x = \frac{\;-\;7\;-\;11}{2},\;\frac{\;-\;7+11}{2}$

$x = \frac{-18}{2},\; \frac{4}{2} = \;-9,\;2$

3. Find the equation of the line if slope $= 6$ and the x-intercept $= 7$.

Solution: 

We know that the general equation of a line with slope m is given by: $y = mx + c$. Using the formula of the x-intercept

$x = \frac{-c}{m}$

$7 = \frac{-c}{6}$

$c = -\;42$

On substituting the value of c and m, we have,

$y = 6x\;-\;42$

4. Find the x-intercept of a line passing through points (2,3) and (4,7)?

Solution: 

The slope of the line passing through the points (2, 3) and (4, 7) is 

$m = \frac{7 \;-\; 3}{4 \;-\; 2} = \frac{4}{2} = 2$

The equation of the line is: $y\;−\;b = m (x\;−\;a)$

$y\;−\;7 = 2(x\;−\;4)$

Substituting $y = 0$, we get

$\;-\;7 = 2(x\;-\;4)$

$\frac{-7}{2} + 4 = x$

$x = \frac{-7 + 8}{2} = \frac{1}{2}$

Practice Problems on X-intercept

Frequently Asked Questions on X-intercept