What is Point-Slope Form

The point-slope form of a line equation is useful when you know the slope of a line and a specific point on the line. This form allows you to quickly write the equation of the line and is particularly helpful in cases where you don’t have the y-intercept. The point-slope form is expressed as:

\( y – y_1 = m(x – x_1) \)

In this formula:

  • \( m \) is the slope of the line
  • \( (x_1, y_1) \) is a point on the line, where \( x_1 \) and \( y_1 \) are the coordinates of that point
  • \( y \) and \( x \) are the variables representing any point on the line

Derivation of the Point-Slope Form

The point-slope form is derived from the concept of slope, which is defined as “rise over run.” If we know a point \( (x_1, y_1) \) on a line and the slope \( m \), we can set up the relationship between any point \( (x, y) \) on the line and this known point:

The slope \( m \) is calculated as \( m = \dfrac{y – y_1}{x – x_1} \).

Rearranging, we get \( y – y_1 = m(x – x_1) \).

This equation, \( y – y_1 = m(x – x_1) \), is known as the point-slope form of the equation of a line.

How to Use the Point-Slope Form

The point-slope form, \( y – y_1 = m(x – x_1) \), is valuable for writing the equation of a line when you know:

  • A point on the line, \( (x_1, y_1) \)
  • The slope \( m \) of the line

You can use this form to quickly write and even rearrange the line’s equation into slope-intercept form if needed.

Example 1: Writing an Equation in Point-Slope Form

Problem: Write the equation of a line that passes through the point \( (3, -2) \) with a slope of \( m = 4 \).

Solution:

  1. Use the point-slope form \( y – y_1 = m(x – x_1) \).
  2. Substitute \( x_1 = 3 \), \( y_1 = -2 \), and \( m = 4 \): \( y – (-2) = 4(x – 3) \).
  3. Simplify the expression: \( y + 2 = 4(x – 3) \).

The equation of the line in point-slope form is \( y + 2 = 4(x – 3) \).

Example 2: Converting Point-Slope Form to Slope-Intercept Form

Problem: Write the equation of a line that passes through \( (1, 5) \) with a slope of \( m = -3 \) and convert it to slope-intercept form.

Solution:

  1. Use the point-slope form \( y – y_1 = m(x – x_1) \).
  2. Substitute \( x_1 = 1 \), \( y_1 = 5 \), and \( m = -3 \): \( y – 5 = -3(x – 1) \).
  3. Distribute \( -3 \): \( y – 5 = -3x + 3 \).
  4. Isolate \( y \) to write in slope-intercept form: \( y = -3x + 8 \).

The equation of the line in slope-intercept form is \( y = -3x + 8 \).

The point-slope form \( y – y_1 = m(x – x_1) \) is a flexible tool for writing line equations, especially when you have a point and a slope, making it easy to rearrange into other forms as needed.