# 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**

**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**

**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

- 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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**Frequently Asked Questions (FAQs)**

## 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.