4.6 Find the Equation of a Line

Linear equations are the foundation of algebraic graphing. They describe straight lines and are crucial for modeling relationships between variables. Understanding how to create and manipulate these equations is key to solving real-world problems and more complex math.

This section covers different methods for writing linear equations. You'll learn how to use slope, intercepts, and points to construct equations. These skills are essential for graphing lines and analyzing their relationships, setting the stage for more advanced algebraic concepts.

Equations of Lines

Line equation from slope and y-intercept

• Slope-intercept form $y = mx + b$ used to construct line equation
  • $m$ slope of the line indicates steepness and direction (positive slopes rise, negative slopes fall)
  • $b$ y-intercept where line crosses y-axis (0, b) provides starting point
• Substitute given slope value for $m$ and y-intercept value for $b$
  • Plug in values to create specific equation for the line
• Example: slope 2, y-intercept -3
  • $y = 2x - 3$ is the equation of the line

Line equation from slope and point

• Point-slope form $y - y_1 = m(x - x_1)$ derives line equation
  • $m$ slope of the line indicates steepness and direction
  • $(x_1, y_1)$ coordinates of a point on the line serve as a reference
• Substitute given slope for $m$ and point coordinates for $x_1$ and $y_1$
  • Plug in values to create specific equation for the line
• Simplify equation by distributing slope and combining like terms
• Example: slope -1/2, point (4, 6)
  • $y - 6 = -\frac{1}{2}(x - 4)$ is the equation of the line

Line equation from two points

• Calculate slope $m$ using slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two given points
  • Slope represents change in y over change in x (rise over run)
• Substitute calculated slope $m$ and coordinates of either point into point-slope form $y - y_1 = m(x - x_1)$
  • Plug in values to create specific equation for the line
• Simplify equation by distributing slope and combining like terms
• Example: points (1, 3) and (5, 11)
  1. $m = \frac{11 - 3}{5 - 1} = 2$
  2. $y - 3 = 2(x - 1)$ is the equation of the line

Equation of parallel line

• Parallel lines have the same slope but different y-intercepts
  • Identical steepness and direction but shifted up or down
• Use the same slope as the given line and a different y-intercept or point to create parallel line equation
  • Maintain slope but adjust the starting point
• Substitute slope and y-intercept or point into slope-intercept form $y = mx + b$ or point-slope form $y - y_1 = m(x - x_1)$
• Example: given line $y = 3x - 2$, parallel line through point (1, 5)
  • $y - 5 = 3(x - 1)$ is the equation of the parallel line

Equation of perpendicular line

• Perpendicular lines have slopes that are negative reciprocals $m_1 = -\frac{1}{m_2}$
  • Product of slopes equals -1, indicating 90° angle between lines
• Find negative reciprocal of given line's slope
  • If given slope is $m$, perpendicular slope is $-\frac{1}{m}$
• Use negative reciprocal slope and given point to create perpendicular line equation with point-slope form $y - y_1 = m(x - x_1)$
• Example: given line $y = -2x + 3$, perpendicular line through point (2, -1)
  1. $m_{perp} = \frac{1}{2}$ is the slope of the perpendicular line
  2. $y - (-1) = \frac{1}{2}(x - 2)$ is the equation of the perpendicular line

Linear Functions and Graphs

• A linear function is a relationship between variables that produces a straight line when graphed on a coordinate plane
• The rate of change in a linear function is constant, represented by the slope
• The graph of a line is a visual representation of all points satisfying a linear equation
• Linear equations can be written in various forms, including slope-intercept and point-slope