뒤로Equations of Lines and Their Properties in College Algebra
스터디 가이드 - 스마트 노트
자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.
Equations of Lines
Standard Form
The standard form of a linear equation is written as , where A, B, and C are integers, A \geq 0, and the greatest common factor (GCF) of A, B, and C is 1. This form is useful for quickly finding x- and y-intercepts and for certain algebraic manipulations.
Standard Form:
Requirements: ; GCF of , , is 1
Application: Useful for graphing and solving systems of equations
Slope-Intercept Form
The slope-intercept form of a line is , where m is the slope and b is the y-intercept. This form allows for easy identification of the slope and y-intercept, making it ideal for graphing.
Slope-Intercept Form:
Slope (m): Measures the steepness of the line
Y-intercept (b): The point where the line crosses the y-axis
Application: Quickly graphing and identifying line properties
Point-Slope Form
The point-slope form is used when the slope and a point on the line are known. It is written as , where is a point on the line and m is the slope.
Point-Slope Form:
Application: Useful for constructing a line from a known point and slope
Finding the Equation of a Line
Given Slope and Y-Intercept
To find the equation of a line with a given slope and y-intercept, substitute the values into the slope-intercept form.
Example: Slope , y-intercept
Equation:
Given Slope and a Point
Use the point-slope form when the slope and a point are given. Substitute the values and simplify to slope-intercept or standard form as needed.
Example: Slope , point
Equation:
Simplified: (slope-intercept form)
Given Two Points
To find the equation of a line passing through two points, first calculate the slope, then use point-slope form.
Slope Formula:
Example: Points and
Slope:
Equation:
Simplified:
Equations of Horizontal and Vertical Lines
Vertical Lines
A vertical line through has the equation . The slope is undefined.
Equation:
Slope: Undefined
Horizontal Lines
A horizontal line through has the equation . The slope is zero.
Equation:
Slope: 0
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines have the same slope and never intersect. To find the equation of a parallel line, use the same slope as the given line and apply point-slope form.
Definition: Lines with equal slopes that do not intersect
Equation: Use the same slope as the original line
Perpendicular Lines
Perpendicular lines intersect at a 90° angle. Their slopes are opposite reciprocals, meaning the product of their slopes is -1.
Definition: Lines with slopes and such that
Equation: If the original slope is , the perpendicular slope is
Summary of Linear Equation Forms
Equation | Description | When to Use |
|---|---|---|
Slope formula | Finding slope between two points | |
Slope-intercept form | Graphing, identifying slope and y-intercept | |
Point-slope form | Known slope and a point | |
Standard form | Finding intercepts, solving systems | |
Horizontal line | Line with slope 0 | |
Vertical line | Line with undefined slope |
Example: Finding Perpendicular Slope
Given , the slope is . The perpendicular slope is .
