Equations of a Line: Point-Slope and Slope-Intercept Forms

Equations of a Line

Point-Slope Form and Slope-Intercept Form

In analytic geometry, the equation of a line can be written in several forms. Two of the most common are the point-slope form and the slope-intercept form. These forms are useful for describing a line given certain conditions, such as passing through specific points.

Key Terms and Definitions

• Slope (m): The measure of the steepness of a line, calculated as the ratio of the change in y to the change in x between two points on the line.

• Point-Slope Form: An equation of a line that uses the slope and a point on the line: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: An equation of a line that uses the slope and the y-intercept: $y = mx + b$

Finding the Equation of a Line Given Two Points

Given two points, $(-4, 9)$ and $(1, -1)$, we can find the equation of the line passing through them.

1. Calculate the Slope (m): The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the given points: $m = \frac{-1 - 9}{1 - (-4)} = \frac{-10}{5} = -2$

2. Write the Point-Slope Form: Using the point $(-4, 9)$ and $m = -2$: $y - 9 = -2(x + 4)$

3. Convert to Slope-Intercept Form: Expand and solve for $y$: $y - 9 = -2(x + 4)$ $y - 9 = -2x - 8$ $y = -2x + 1$

Summary Table: Forms of the Line Equation

Form	Equation	Parameters
Point-Slope	$y - 9 = -2(x + 4)$	Point $(-4, 9)$, Slope $-2$
Slope-Intercept	$y = -2x + 1$	Slope $-2$, Intercept $1$

Example

• Given: Points $(-4, 9)$ and $(1, -1)$

• Find: Equation of the line in point-slope and slope-intercept forms

• Solution:

  • Slope: $m = -2$

  • Point-Slope Form: $y - 9 = -2(x + 4)$

  • Slope-Intercept Form: $y = -2x + 1$