# x intercept

x intercept

Tim and Janne were observing various graphs. They observed that some crossed the x-axis, while others didn't. They were curious to know more about the curves and lines that cross the x-axis.

Can we say that x-intercept is the value of the $$x$$ coordinate of a point where the value of $$y$$ coordinate equal to zero?

Let’s learn about x-intercept, x-intercept in an equation, x-intercept formula, x-intercept definition, and x-intercept example, and explore x-intercept calculator for better understanding.

Check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

## Lesson Plan

 1 What is x-Intercept? 2 Tips and Tricks 3 Important Notes on x-Intercept 4 Solved Examples on x-Intercept 5 Interactive Questions on x-Intercept

## What is x-Intercept?

The x-intercept for any curve is the value of x coordinate of the point where the graph cuts x-axis, or we can say that x-intercept is the value of the x coordinate of a point where the value of y coordinate equal to zero.

## General Formulae for X-Intercept

### Staright Line

The general form of a straight line is

 $$ax + by +c = 0$$

where $$a, b, c$$ are constants

x-intercept of the line can be obtained by putting $$y=0$$

 $$x \text{-intercept} = \dfrac{-c}{a}$$

### Slope Intercept Line

Slope-intercept form of a straight line is

 $$y = mx + c$$

where $$m$$ is the slope of the line
$$c$$ is the y-intercept

The x-intercept of the line can be obtained by putting $$y=0$$

 $$x \text{-intercept}= \dfrac{-c}{m}$$

### Point Slope Line

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 $$y=0$$

 $$x \text{-intercept}= \dfrac{am -b}{m}$$

## How to Find x-Intercept of a Straight Line?

To find the x-intercept of a line of the form $$y= mx +b$$,
substitute y = 0

### Example

To find the x-intercept of any line for example the line $$y = 2x -4$$, put $$y=0$$ in the equation of line.

### Solution

\begin{align} 0 &= 2x - 4\\[0.2cm] 4 &= 2x\\[0.2cm] 2 &= x\\[0.2cm] \end{align}

So, the x-intercept of the line $$y = 2x -4$$ is $$2$$

Similarly, by substituting x = 0, we can find the y-intercept

Let's see an x-intercept example.

### Example

John wants to find the equation of a line with the slope equal to 2 and the x-intercept equal to -5. Can you help him?

### Solution

The general equation of a line with slope m is $$y = mx + c$$
The x-intercept of the line is $$-5$$
$\text{using the formula for x-intercept}$
\begin{align} x &= \dfrac{-c}{m} \\[0.2cm] -5 &= \dfrac{-c}{2} \\[0.2cm] 10 &= c \\[0.2cm] \end{align}
$\text{Put the value of c and m}$
$y = 2x + 10$

Equation of the line is $$y=2x+10$$

### Intercept Calculator

Tips and Tricks
• If the x-intercept and the y-intercept of a line are equal, then the line forms a 45 degree angle with the x-axis.
• To find the x-intercept of any line ax+by+c=0, put the value of y as 0
• To find the point of intersection of two lines, solve the two equations simultaneously.
• For two perpendicular lines, the product of their slopes is -1

## How to Find x-Intercept of a Quadratic Function or a Parabola?

The x-intercept of any curve can be obtained using a similar technique. We just have to put the value of y as 0 in the equation of the curve.

The general equation of a parabola is $$y = ax^2 + bx + c$$. We can calculate x-intercept by puting y as 0

### Example

Find x-intercept of the parabola $$y = x^2 - 3x +2$$.

### Solution

Put $$y=0$$ in the equation of parabola.

$0 = x^2 - 3x +2$
We can apply regular methods to solve a quadratic equation and calculate the value of x-intercepts.
$$x = 2 \text{ and } x = 1$$ are the two roots of the equation.
It means that the graph is crossing x-axis at two different points.

So, $$x = 2 \text{ and } x = 1$$ are the two x-intercepts of the parabola

Similarly we can calculate y-intercept by putting x = 0

Important Notes
• The x-intercept of a line is the distance of the x coordinate from the point where the line cuts the x-axis from the origin.
• The x-intercept is a point on the graph where y is zero.
• If a line is parallel to the y-axis, its x-intercept is not defined.
• For any curve, the x-intercepts of the curve = roots of the equation = solution of the equation = zeros of the curve.

## Solved Examples

 Example 1

Shawn says that the line $$2y = 3x - 6$$ has x-intercept equal to $$-3$$ and y-intercept equal to $$2$$ while Judy says that the line has x-intercept equal to $$2$$ and y-intercept equal to $$-3$$.
Who is right?

Solution

Given the line $$2y = 3x - 6$$
x-intercept can be calculated by putting y = 0
\begin{align} 2 \times 0 &= 3x -6\\[0.2cm] 0 &= 3x -6\\[0.2cm] 6 &= 3x\\[0.2cm] 2 &= x \\[0.2cm] \text{x intecept of the line is 2}\\[0.2cm] \end{align}
y-intercept can be calculated by putting x = 0
\begin{align} 2y &= 3 \times 0 -6\\[0.2cm] 2y &= 0 -6\\[0.2cm] 2y &= -6\\[0.2cm] y &= -3\\[0.2cm] \text{y-intercept of the line is -3} \\[0.2cm] \end{align}

 $$\therefore$$ Judy is right.
 Example 2

Mathew draw a line with slope -2  which passes through point $$P(a, 1)$$. If the sum of both the intercepts of the line is equal to 6, find the value of $$a$$.

Solution

General equation for a line with a given slope and a point is
$y - b = m(x - a)$

Where $$(a, b)$$ is the point on the line

\begin{align} y - 1 &= -2(x - a)\\[0.2cm] \text{x-intercept} &= \dfrac{1}{2} + a\\[0.2cm] \text{y-intercept} &= 1 + 2a\\[0.2cm] \text{sum of intercepts} &= 1 + 2a + \dfrac{1}{2} + a\\[0.2cm] \text{sum of intercepts} &= 3a + \dfrac{3}{2}\\[0.2cm] 6 &= 3a + \dfrac{3}{2}\\[0.2cm] \dfrac{9}{2} &= 3a\\[0.2cm] \dfrac{3}{2} &= a\\[0.2cm] \end{align}

 $$\therefore$$ Value of $$a$$ is $$\dfrac{3}{2}$$
 Example 3

Find the x and y-intercepts of the curve $$9x^2 + 16y^2 = 25$$

Solution

x-intercept of the curve can be calculated by putting $$y =0$$
\begin{align} 9x^2 + 16 \times 0^2 &= 25\\[0.2cm] 9x^2 + 0 &= 25\\[0.2cm] x^2 &= \dfrac{25}{9}\\[0.2cm] x^2 &= (\dfrac{5}{3})^2\\[0.2cm] x &= \pm\dfrac{5}{3}\\[0.2cm] \end{align}
y-intercept of the curve can be calculated by putting $$x =0$$
\begin{align} 9 \times 0^2 + 16y^2 &= 25\\[0.2cm] 0 + 16y^2 &= 25\\[0.2cm] y^2 &= \dfrac{25}{16}\\[0.2cm] y^2 &= (\dfrac{5}{4})^2\\[0.2cm] y &= \pm\dfrac{5}{4}\\[0.2cm] \end{align}

 $$\therefore$$ x-intercepts are $$\dfrac{5}{3} \text{ and} -\dfrac{5}{3}$$ y-intercepts are $$\dfrac{5}{4} \text{ and} -\dfrac{5}{4}$$

## Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

We hope you enjoyed learning about x-intercept with the simulations and practice questions. Now you will be able to easily solve problems on x-intercept in an equation, x-intercept formula, x-intercept definition, and x-intercept example.

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. Can 0 be an x-intercept?

Yes, 0 can be an x-intercept for the line $$y = mx$$, where m is the slope of the line.

## 2. What is the y-intercept formula?

y-intercept formula for a line $$y = mx + c$$ is $$y = c$$.

## 3. How do you find x in an equation?

We can find $$x$$ by putting the value of $$y = 0$$ in any equation. The value of $$x$$ so obtained is the x-intercept of the curve.

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