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Slope and Linear Equations
Rise over Run, Slope-Intercept, and Point-Slope Forms
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Slope measures how steeply a line rises or falls. It is defined as the ratio of vertical change (rise) to horizontal change (run) between any two points on the line: m = (y₂ − y₁)/(x₂ − x₁). A positive slope means the line rises left to right; a negative slope means it falls; slope = 0 means horizontal; an undefined slope means vertical (division by zero). Because a line has constant slope everywhere, you only need two points to calculate it.
The slope-intercept form y = mx + b is the most common way to write a linear equation. Here m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis). This form makes graphing straightforward: start at (0, b) and use the slope to find additional points. Point-slope form y − y₁ = m(x − x₁) is useful when you know a point and the slope but not the y-intercept. Two lines with the same slope are parallel; perpendicular lines have slopes that are negative reciprocals of each other (m₁ × m₂ = −1).
Key Facts
- Slope: m = (y₂ − y₁)/(x₂ − x₁) = rise/run
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
- Point-slope form: y − y₁ = m(x − x₁)
- Standard form: Ax + By = C (useful for x and y intercepts)
- Parallel lines: same slope (m₁ = m₂); perpendicular lines: negative reciprocal slopes (m₁ × m₂ = −1)
- Horizontal line: slope = 0, equation y = k; vertical line: undefined slope, equation x = k
Vocabulary
- Slope
- The ratio of vertical change to horizontal change between two points on a line; measures steepness and direction.
- Y-intercept
- The point where a line crosses the y-axis (where x = 0); the value b in y = mx + b.
- Slope-intercept form
- The equation form y = mx + b, where m is the slope and b is the y-intercept; directly shows the rate of change and starting value.
- Linear equation
- An equation whose graph is a straight line; all variables are to the first power and no variable is multiplied by another.
- Rate of change
- How much one quantity changes per unit change in another; for a linear function, this is constant and equal to the slope.
Common Mistakes to Avoid
- Computing slope as run/rise instead of rise/run. Slope = change in y divided by change in x (vertical over horizontal). Reversing this gives the reciprocal, not the slope.
- Confusing the y-intercept with the x-intercept. The y-intercept (b in y = mx + b) is where the line crosses the y-axis (set x = 0). The x-intercept is where it crosses the x-axis (set y = 0 and solve for x).
- Assuming any two points give a different slope for the same line. Slope is constant everywhere on a straight line — it doesn't matter which two points you choose.
- Misapplying the perpendicular slope rule. Perpendicular slopes are negative reciprocals: if m = 2/3, the perpendicular slope is −3/2. Both the sign change and the flip are required.
Practice Questions
- 1 Find the slope and y-intercept of the line passing through (−2, 5) and (4, −1). Write the equation in slope-intercept form.
- 2 Line A has equation y = (3/4)x − 2. Write the equation of a line perpendicular to A passing through the point (0, 5).
- 3 A car travels at a constant speed. After 2 hours it has gone 120 km; after 5 hours it has gone 300 km. What is the slope of the distance-time graph and what does it represent physically?