Overview
This lecture explains how to find the slope and y-intercept of a linear equation, focusing on manipulating equations into slope-intercept form (y = mx + b).
Slope-Intercept Form Basics
- Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
- The slope (m) is the coefficient of x; the y-intercept (b) is the constant term.
Finding Slope and Y-Intercept from Equations
- For y = 3/4x - 5, slope m = 3/4, y-intercept b = -5.
- For y = 8 - 4x, rewrite as y = -4x + 8; slope m = -4, y-intercept b = 8.
- For y = 5 - x, rewrite as y = -x + 5; slope m = -1, y-intercept b = 5.
- For y = -7x, rewrite as y = -7x + 0; slope m = -7, y-intercept b = 0.
Special Cases: Horizontal and Vertical Lines
- For y = 3, rewrite as y = 0x + 3; slope m = 0, y-intercept b = 3 (horizontal line).
- For x = 4, vertical line; slope is undefined, no y-intercept.
Calculating Slope with Two Points
- Slope formula: m = (y₂ - y₁) / (x₂ - x₁).
- Horizontal lines: m = 0 (no rise).
- Vertical lines: denominator is zero, so slope is undefined.
Positive and Negative Examples
- For y = -6, slope m = 0, y-intercept b = -6 (horizontal line).
- For x = -2, slope is undefined, no y-intercept (vertical line).
Converting Standard Form to Slope-Intercept Form
- Standard form: Ax + By = C.
- Solve for y: isolate y on one side to write in y = mx + b form.
- Example: -2x + 2y = 6 → y = 2x + 3, so m = 2, b = 3.
- Example: 3x - 5y = 8 → y = (3/5)x - 8/5, so m = 3/5, b = -8/5.
- If fractions present, clear them first (e.g., multiply both sides by denominator).
Key Terms & Definitions
- Slope (m) — The rate at which y changes with respect to x; coefficient of x in y = mx + b.
- Y-intercept (b) — The value of y where the line crosses the y-axis (when x = 0).
- Slope-intercept form — Linear equation written as y = mx + b.
- Standard form — Linear equation written as Ax + By = C.
- Horizontal line — Line with equation y = constant; slope is 0.
- Vertical line — Line with equation x = constant; slope is undefined.
Action Items / Next Steps
- Practice converting linear equations to slope-intercept form and identifying m and b.
- Try additional problems involving standard form equations.
- Review the slope formula and apply it to given points.