Equations of Lines

Slope-Intercept Form

The slope-intercept form of a linear equation expresses the relationship between the variables x and y in the form , where m is the slope and b is the y-intercept. This form is widely used for graphing and analyzing linear functions.

• Slope (m): Measures the steepness of the line; calculated as the change in y divided by the change in x between two points.

• Y-intercept (b): The point where the line crosses the y-axis, represented as (0, b).

• Equation:

• Example: Find the equation for the line containing (0, 5) and (2, 1): Slope: Y-intercept: Equation:

Point-Slope Form

The point-slope form is useful when the slope and a specific point on the line are known. It is written as , where is a point on the line.

• Equation:

• Example: Find the point-slope form for the line passing through (1, 3) and (4, -2): Slope: Point: Equation:

Converting Point-Slope to Slope-Intercept Form

To convert from point-slope to slope-intercept form, expand and simplify the equation.

• Example: Line through (1, 3) and (4, 9): Slope: Point-slope form: Expand: Slope-intercept form:

Parallel Lines

Nonvertical parallel lines have equal slopes. To find the equation of a parallel line, use the same slope as the given line and pass it through the specified point.

• Key Property: Parallel lines:

• Example: Find the equation of the line parallel to passing through (2, 4): Slope: Point: Point-slope form: Expand: Slope-intercept form:

• Example: Parallel to through (3, -1): Convert to slope-intercept: Slope: Point-slope form: Expand:

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals (inverse opposites) of each other. If the slope of one line is , the slope of the perpendicular line is .

• Key Property: Perpendicular slopes:

• Example: Perpendicular to through (4, 3): Slope of given line: Perpendicular slope: Point-slope form: Expand:

• Example: Perpendicular to through (8, 5): Convert to slope-intercept: Perpendicular slope: Point-slope form: Expand:

Intercepts of a Line

The x-intercept is the point where the line crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0).

• Finding x-intercept: Set and solve for .

• Finding y-intercept: Set and solve for .

• Example: For : x-intercept: (Point: (3, 0)) y-intercept: (Point: (0, -5))

Applications: Linear Models

Linear equations can model real-world situations, such as predicting life expectancy over time. The slope represents the rate of change, and the y-intercept represents the initial value.

• Example: Life expectancy data: 1990: 59.6 years (t = 0) 1998: 63.2 years (t = 8) Slope: Y-intercept: Linear function: Predict for 2010 (t = 20): Interpretation: The model predicts a life expectancy of 69.6 years in 2010.

Summary Table: Forms of Linear Equations

Additional info: The notes have been expanded to include definitions, step-by-step examples, and a summary table for clarity and completeness.