Slopes and Equations of Lines: Study Notes for College Algebra

Slopes and Equations of Lines

Finding the Slope Between Two Points

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y (vertical change) to the change in x (horizontal change) between two points on the line.

• Definition: The slope m between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

• Example: Find the slope between (5, 13) and (20, -6): $m = \frac{-6 - 13}{20 - 5} = \frac{-19}{15}$

Finding the Slope from a Linear Equation

Linear equations can be written in various forms. The slope-intercept form is especially useful for identifying the slope directly:

• Slope-Intercept Form: $y = mx + b$, where m is the slope and b is the y-intercept.

• To find the slope from equations not in this form, rearrange the equation to solve for y.

• Example: For $5x - 4y = 4$: Rearranged: $-4y = -5x + 4$ $y = \frac{5}{4}x - 1$ Slope: $m = \frac{5}{4}$

Slopes of Parallel and Perpendicular Lines

Understanding the relationship between the slopes of parallel and perpendicular lines is essential in analytic geometry.

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Have slopes that are negative reciprocals of each other. If the slope of one line is m, the slope of a perpendicular line is $-\frac{1}{m}$ (provided $m \neq 0$).

• Example: If a line has slope $\frac{3}{2}$, a perpendicular line has slope $-\frac{2}{3}$.

Point-Slope Form of a Line

The point-slope form is useful for writing the equation of a line when you know its slope and a point on the line.

• Formula: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and m is the slope.

• Example: Through (0, 2), parallel to $y = -5$ (which is a horizontal line, so $m = 0$): $y - 2 = 0(x - 0)$ $y = 2$

Writing Equations Given Slope and Y-Intercept

When the slope and y-intercept are known, use the slope-intercept form:

• Formula: $y = mx + b$

• Example: Slope = -1, y-intercept = 3: $y = -1x + 3$

Summary Table: Forms of Linear Equations

Form	General Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope (m), y-intercept (b)
Point-Slope	$y - y_1 = m(x - x_1)$	Point $(x_1, y_1)$, Slope (m)
Standard	$Ax + By = C$	Can be rearranged to find slope and intercepts

Key Terms

• Slope (m): The ratio of vertical change to horizontal change between two points on a line.

• Y-intercept (b): The point where the line crosses the y-axis.

• Parallel Lines: Lines with the same slope.

• Perpendicular Lines: Lines whose slopes are negative reciprocals.