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

Chapter 3: Equations and Inequalities in Two Variables and Functions

Section 3.3: Graphs and the Equations of a Line

This section introduces the fundamental concepts of linear equations in two variables, focusing on their graphical representations and algebraic forms. Understanding these forms is essential for analyzing and interpreting linear relationships in College Algebra.

Slope-Intercept Form

The slope-intercept form of a linear equation is a widely used method for expressing the equation of a line. It is given by:

• General Form: $y = mx + b$

• Where:

  • m is the slope of the line (rate of change of y with respect to x).

  • b is the y-intercept (the value of y when x = 0).

Key Properties:

• The slope, m, determines the steepness and direction of the line.

• The y-intercept, b, is the point where the line crosses the y-axis.

Examples:

• For $y = 3x - 4$:

  • Slope: 3

  • Y-intercept: (0, -4)

• For $y = \frac{3}{8}x + \frac{1}{4}$:

  • Slope: $\frac{3}{8}$

  • Y-intercept: (0, $\frac{1}{4}$)

Constructing the Equation of a Line

To write the equation of a line, you need the slope and the y-intercept.

• Example: Find the equation of the line with slope $\frac{2}{3}$ and y-intercept (0, -5):

  • $y = mx + b$

  • $y = \frac{2}{3}x + (-5)$

  • $y = \frac{2}{3}x - 5$

Finding the Equation from a Graph

Given a graph, you can determine the slope and y-intercept by analyzing the changes in x and y.

• Example: Given points (0, 2) and (3, 0):

  • Y-intercept: 2

  • Slope: $\frac{\text{change in } y}{\text{change in } x} = \frac{-2}{3}$

  • Equation: $y = -\frac{2}{3}x + 2$

Converting to Slope-Intercept Form

Any linear equation can be rearranged into slope-intercept form to easily identify the slope and y-intercept.

• Example: Convert $5x + 3y = 6$ to slope-intercept form:

  • $3y = -5x + 6$

  • $y = -\frac{5}{3}x + 2$

  • Slope: $-\frac{5}{3}$

  • Y-intercept: (0, 2)

Point-Slope Form

The point-slope form is useful when you know the slope and a specific point on the line. It is given by:

• General Form: $y - y_1 = m(x - x_1)$

• Where:

  • m is the slope

  • $(x_1, y_1)$ is a known point on the line

Example: Find the equation of the line passing through (4, -3) with slope 5, in standard form:

• $y - (-3) = 5(x - 4)$

• $y + 3 = 5x - 20$

• $y = 5x - 23$

• $5x - y = 23$ (Standard form)

Finding the Equation from Two Points

When given two points, you can calculate the slope and use point-slope form to find the equation.

• Formula for Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Points (2, 1) and (7, 4):

  • $m = \frac{4 - 1}{7 - 2} = \frac{3}{5}$

  • Using point (2, 1): $y - 1 = \frac{3}{5}(x - 2)$

  • Convert to slope-intercept form: $y = \frac{3}{5}x - \frac{6}{5} + 1 = \frac{3}{5}x - \frac{1}{5}$

Parallel and Perpendicular Lines

Lines that are parallel have the same slope. Lines that are perpendicular have slopes that are negative reciprocals.

• Parallel Lines:

  • If a line has slope m, any line parallel to it also has slope m.

  • Example: Find the equation of the line passing through (1, 0) and parallel to $3x - y = 4$:

    • First, write $3x - y = 4$ in slope-intercept form: $y = 3x - 4$

    • Slope is 3.

    • Use point-slope form: $y - 0 = 3(x - 1)$

    • $y = 3x - 3$

    • Standard form: $3x - y = 3$

• Perpendicular Lines:

  • If a line has slope m, a perpendicular line has slope $-\frac{1}{m}$.

  • Example: Find the equation of the line passing through (6, 2) and perpendicular to $2x + y = 8$:

    • First, write $2x + y = 8$ in slope-intercept form: $y = -2x + 8$

    • Slope is -2, so perpendicular slope is $\frac{1}{2}$.

    • Use point-slope form: $y - 2 = \frac{1}{2}(x - 6)$

    • $y - 2 = \frac{1}{2}x - 3$

    • $y = \frac{1}{2}x - 1$

    • Standard form: $x - 2y = 2$

Summary Table: Forms of Linear Equations

Form	Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope (m), y-intercept (b)
Point-Slope	$y - y_1 = m(x - x_1)$	Slope (m), point $(x_1, y_1)$
Standard	$Ax + By = C$	General linear form

Additional info: The notes are based on textbook slides for Intermediate/College Algebra, focusing on Section 3.3. All examples and forms are standard for introductory college-level algebra courses.