What are X and Y intercepts?

The X-intercept of a function is the point where the graph crosses the X-axis. It occurs when \( y = 0 \).

The Y-intercept of a function is the point where the graph crosses the Y-axis. It occurs when \( x = 0 \).

Formal Definition: A point \( (x, y) \) is the intercept if either \( x = 0 \) for the Y-intercept or \( y = 0 \) for the X-intercept.

\[ x\text{-intercept:} \quad y = 0 \] \[ y\text{-intercept:} \quad x = 0 \]

How to find X and Y intercept

How to Find X-Intercepts and Y-Intercepts?

To find the X-intercept, set \( y = 0 \) in the equation and solve for \( x \).
To find the Y-intercept, set \( x = 0 \) in the equation and solve for \( y \).

\[ \text{For the X-intercept:} \quad y = 0, \quad \text{solve for} \ x \] \[ \text{For the Y-intercept:} \quad x = 0, \quad \text{solve for} \ y \]

Intercept form of line

Intercept Form of a Line

The intercept form of a linear equation is given by:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

Where \( a \) is the X-intercept and \( b \) is the Y-intercept of the line.

The Slope Intercept form

Finding Intercept Form - The Slope-Intercept Form of Lines

The Slope-Intercept Form of a line is given by:

\[ y = mx + b \]

Where \( m \) is the slope of the line, and \( b \) is the Y-intercept. To find the intercept form from the slope-intercept form, rewrite the equation in intercept form by finding the X and Y intercepts.

Point-Slope form

Finding Intercept Form - The Point-Slope Form of Lines

The Point-Slope Form of a line is given by:

\[ y - y_1 = m(x - x_1) \]

Where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. To find the intercept form from the point-slope form, use the slope \( m \) and the given point to calculate the intercepts.

The Two Point form of line

Finding Intercept Form - The Two Point Form of Line

The Two Point Form of a line is given by:

\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} \]

Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line. To find the intercept form from the two-point form, calculate the intercepts using the coordinates of the points.

Solved Examples

Example: Find the X-intercept and Y-intercept of the line equation \( 3x + 4y = 12 \).

Solution:

For the X-intercept: Set \( y = 0 \) in the equation:

\[ 3x + 4(0) = 12 \implies 3x = 12 \implies x = \frac{12}{3} = 4 \]

So, the X-intercept is \( (4, 0) \).

For the Y-intercept: Set \( x = 0 \) in the equation:

\[ 3(0) + 4y = 12 \implies 4y = 12 \implies y = \frac{12}{4} = 3 \]

So, the Y-intercept is \( (0, 3) \).


Example: Find the X-intercept and Y-intercept of the line equation \( 2x - 5y = 10 \).

Solution:

For the X-intercept: Set \( y = 0 \) in the equation:

\[ 2x - 5(0) = 10 \implies 2x = 10 \implies x = \frac{10}{2} = 5 \]

So, the X-intercept is \( (5, 0) \).

For the Y-intercept: Set \( x = 0 \) in the equation:

\[ 2(0) - 5y = 10 \implies -5y = 10 \implies y = \frac{10}{-5} = -2 \]

So, the Y-intercept is \( (0, -2) \).


Example: Write the equation of the line in intercept form that passes through the X-intercept \( (6, 0) \) and the Y-intercept \( (0, 4) \).

Solution:

The intercept form of a line is:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

Where \( a \) is the X-intercept and \( b \) is the Y-intercept. Substituting \( a = 6 \) and \( b = 4 \):

\[ \frac{x}{6} + \frac{y}{4} = 1 \]

This is the equation of the line in intercept form.


Example: A line passes through the points \( (2, 5) \), \( (p, 0) \), and \( (0, q) \), where \( p \) is the X-intercept and \( q \) is the Y-intercept of the line. If the slope of the line is \( \frac{3}{4} \), find the values of \( p \), \( q \), and the equation of the line.

Solution:

The formula for the slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Given that the slope \( m = \frac{3}{4} \), we can use the points \( (2, 5) \) and \( (p, 0) \) to find the relationship between \( p \) and \( q \).

Step 1: Finding the value of \( p \)

Using the slope formula between the points \( (2, 5) \) and \( (p, 0) \):

\[ \frac{3}{4} = \frac{0 - 5}{p - 2} \]

Simplifying:

\[ \frac{3}{4} = \frac{-5}{p - 2} \]

Multiply both sides by \( 4(p - 2) \):

\[ 3(p - 2) = -20 \]

Expanding:

\[ 3p - 6 = -20 \]

Solving for \( p \):

\[ 3p = -14 \quad \Rightarrow \quad p = -\frac{14}{3} \]

Step 2: Finding the value of \( q \)

Now use the points \( (p, 0) \) and \( (0, q) \). The slope between these two points is \( \frac{3}{4} \):

\[ \frac{3}{4} = \frac{q}{\frac{14}{3}} \]

Simplifying:

\[ \frac{3}{4} = \frac{3q}{14} \]

Multiplying both sides by 14:

\[ \frac{3 \times 14}{4} = 3q \quad \Rightarrow \quad q = \frac{7}{2} \]

Step 3: Equation of the line

We can now use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

Using the point \( (2, 5) \) and slope \( m = \frac{3}{4} \), the equation of the line becomes:

\[ y - 5 = \frac{3}{4}(x - 2) \]

Expanding:

\[ y - 5 = \frac{3}{4}x - \frac{3}{2} \]

Adding 5 to both sides:

\[ y = \frac{3}{4}x - \frac{3}{2} + 5 \]

Simplifying:

\[ y = \frac{3}{4}x + \frac{7}{2} \]

Thus, the equation of the line is:

\[ y = \frac{3}{4}x + \frac{7}{2} \]