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

Equations of a Line

Point-Slope Form and Slope-Intercept Form

When given two points on a line, you can write the equation of the line in both point-slope and slope-intercept forms. These forms are fundamental in Precalculus for describing linear relationships.

Key Terms and Definitions

• Point-Slope Form: An equation of a line using a point $(x_1, y_1)$ and the slope $m$:

$y - y_1 = m(x - x_1)$

• Slope-Intercept Form: An equation of a line with slope $m$ and $y$-intercept $b$:

$y = mx + b$

Finding the Slope

• Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• For the points $(-4, -1)$ and $(4, 11)$:

$m = \frac{11 - (-1)}{4 - (-4)} = \frac{12}{8} = \frac{3}{2}$

Writing the Equation in Point-Slope Form

• Using the point $(-4, -1)$ and the slope $\frac{3}{2}$:

$y - (-1) = \frac{3}{2}(x - (-4))$

Simplifies to:

$y + 1 = \frac{3}{2}(x + 4)$

Writing the Equation in Slope-Intercept Form

• Start from the point-slope form and solve for $y$:

$y + 1 = \frac{3}{2}(x + 4)$

$y + 1 = \frac{3}{2}x + 6$

$y = \frac{3}{2}x + 5$

Summary Table: Forms of the Line

Form	Equation
Point-Slope	$y + 1 = \frac{3}{2}(x + 4)$
Slope-Intercept	$y = \frac{3}{2}x + 5$

Example Application

• Given: Points $(-4, -1)$ and $(4, 11)$

• Find: Equation of the line in both forms

• Solution:

  • Calculate slope: $m = \frac{3}{2}$

  • Point-slope form: $y + 1 = \frac{3}{2}(x + 4)$

  • Slope-intercept form: $y = \frac{3}{2}x + 5$