Lines in the Plane: Slope, Equations, and Relationships

Chapter 0: Prerequisites

Section 0.4: Lines in the Plane

This section introduces the fundamental concepts related to lines in the coordinate plane, including slope, various forms of linear equations, and the relationships between parallel and perpendicular lines. Mastery of these concepts is essential for understanding more advanced topics in algebra and precalculus.

Slope of a Line

The slope of a line measures its steepness and direction. It is a key characteristic of linear equations and is used to describe how one variable changes with respect to another.

• Definition: The slope m of a nonvertical line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

• If the line is vertical ($x_1 = x_2$), the slope is undefined.

Example: Find the slope of the line through (3, -2) and (0, 1):

$m = \frac{1 - (-2)}{0 - 3} = \frac{3}{-3} = -1$

The slope is -1.

Point-Slope Form of an Equation of a Line

The point-slope form is useful for writing the equation of a line when you know the 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: Find the equation of the line through (6, 3) with slope 7:

$y - 3 = 7(x - 6)$

Through (9, -3) with slope -1:

$y + 3 = -1(x - 9)$

Slope-Intercept Form of an Equation of a Line

The slope-intercept form is commonly used for graphing and quickly identifying the slope and y-intercept of a line.

• Formula:

$y = mx + b$

• Where $m$ is the slope and $b$ is the y-intercept (the value of $y$ when $x = 0$).

Example: Find the equation of the line through (0, 6) with slope -4:

$y = -4x + 6$

Find the equation through (-5, 4) and (5, 2):

$m = \frac{2 - 4}{5 - (-5)} = \frac{-2}{10} = -\frac{1}{5}$ Using point-slope form: $y - 4 = -\frac{1}{5}(x + 5)$ Simplifying to slope-intercept form: $y = -\frac{1}{5}x + 3$

Forms of Equations of Lines

There are several standard forms for the equation of a line, each useful in different contexts:

Form	Equation	Description
General Form	$Ax + By + C = 0$	$A$ and $B$ not both zero
Slope-Intercept Form	$y = mx + b$	Slope $m$, y-intercept $b$
Point-Slope Form	$y - y_1 = m(x - x_1)$	Point $(x_1, y_1)$, slope $m$
Vertical Line	$x = a$	All points with $x = a$
Horizontal Line	$y = b$	All points with $y = b$

Example: Find the general-form equation for the line through (-5, -1) and (3, 7):

$m = \frac{7 - (-1)}{3 - (-5)} = \frac{8}{8} = 1$ Using point-slope: $y + 1 = 1(x + 5)$ Rearranged: $-x + y - 6 = 0$

Parallel and Perpendicular Lines

Understanding the relationships between lines is crucial for solving geometric and algebraic problems.

• Parallel Lines: Two nonvertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$). Any two distinct vertical lines are also parallel.

• Perpendicular Lines: Two nonvertical lines are perpendicular if and only if their slopes are opposite reciprocals:

$m_1 = -\frac{1}{m_2}$

• A vertical line is perpendicular to a horizontal line, and vice versa.

Example: Find the equation of a line through (2, -3) parallel to $4x + 5y = 10$:

Rewrite $4x + 5y = 10$ in slope-intercept form: $5y = -4x + 10 \implies y = -\frac{4}{5}x + 2$ Slope is $-\frac{4}{5}$ Use point-slope: $y + 3 = -\frac{4}{5}(x - 2)$ Simplify: $y = -\frac{4}{5}x + \frac{8}{5} - 3$

Example: Find the equation of a line through $P(-4, 5)$ perpendicular to $2x - y = 1$:

Rewrite $2x - y = 1$ as $y = 2x - 1$ (slope $m = 2$) Perpendicular slope is $-\frac{1}{2}$ Use point-slope: $y - 5 = -\frac{1}{2}(x + 4)$ Simplify: $y = -\frac{1}{2}x - 2 + 5 = -\frac{1}{2}x + 3$

Summary Table: Forms and Relationships of Lines

Concept	Key Property	Equation/Condition
Slope	Steepness of line	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Parallel Lines	Equal slopes	$m_1 = m_2$
Perpendicular Lines	Opposite reciprocal slopes	$m_1 = -\frac{1}{m_2}$
Vertical Line	Undefined slope	$x = a$
Horizontal Line	Zero slope	$y = b$

Applications

• Linear equations are widely used in business, behavioral science, and many other fields to model relationships between variables.

• Understanding the forms and relationships of lines is foundational for graphing, solving systems, and analyzing data.