Calculate the slope of a line between two points using the slope formula
Basic: (2, 3) and (5, 9)
Origin: (0, 0) and (4, 2)
Horizontal: (1, 5) and (7, 5)
Vertical: (3, 2) and (3, 8)
Negative: (1, 8) and (5, 2)
Decimals: (-2.5, 4.5) and (3.5, -1.5)
The slope of a line between two points (x₁, y₁) and (x₂, y₂) is:
The slope represents the rate of change: how much y changes for each unit change in x.
Positive slope: Line rises from left to right
Negative slope: Line falls from left to right
Zero slope: Horizontal line (y = constant)
Undefined slope: Vertical line (x = constant)
Calculate road steepness and inclines for highway design. A 5% grade means 5 feet of elevation change per 100 feet horizontally.
Analyze price trends, revenue growth rates, and cost changes over time using slope as the rate of change indicator.
Determine medication absorption rates and concentration changes in bloodstream over time intervals.
Track athlete improvement rates, speed changes during races, and training progress over time periods.
Calculate bearing angles for GPS navigation, drone flight paths, and maritime route planning.
Swapping coordinates inconsistently
Use (y₂ - y₁) / (x₂ - x₁), not (y₂ - y₁) / (x₁ - x₂). Keep the same order for both numerator and denominator.
Confusing rise and run
Rise is the vertical change (Δy), run is horizontal (Δx). Slope = rise/run, not run/rise.
Saying vertical slope is zero
Vertical lines have undefined slope (division by zero), not zero. Horizontal lines have zero slope.
Misinterpreting negative slopes
Negative slope doesn't mean "wrong" - it just means the line decreases from left to right (downhill).
Best Practice
Label your points clearly as (x₁, y₁) and (x₂, y₂) before calculating. Draw a sketch to visualize rise and run.
| Form | Equation | Best Used For | Key Feature |
|---|---|---|---|
| Slope-Intercept | Graphing; identifying slope and y-intercept quickly | Shows slope (m) and y-intercept (b) directly | |
| Point-Slope | Writing equations when you know slope and one point | Easy to write from given information | |
| Standard Form | Finding x and y intercepts; integer coefficients | Symmetrical; good for linear systems | |
| Two-Point Form | Writing equations from two points directly | No need to calculate slope first |
Understanding the relationship between slopes helps solve geometric problems and analyze line relationships.
Parallel lines have equal slopes. They never intersect and maintain constant distance.
Example: If line 1 has slope 3, any parallel line also has slope 3.
Perpendicular lines have negative reciprocal slopes. They intersect at 90° angles.
Example: If line 1 has slope 2/3, perpendicular line has slope -3/2.
Special Case: Horizontal lines (slope = 0) are perpendicular to vertical lines (undefined slope). The negative reciprocal rule doesn't apply to vertical/horizontal pairs.
Slope isn't just an abstract math concept — it's literally written into federal law. The Americans with Disabilities Act (ADA) requires every public building to have wheelchair-accessible ramps that meet exact slope specifications.
This means for every 1 inch of vertical rise, the ramp must extend at least 12 inches horizontally.
A building entrance is 30 inches above ground. What's the minimum ramp length?
= 30 feet of ramp! That's why many ramps switchback.
The side-to-side slope (perpendicular to travel direction) must not exceed:
This prevents wheelchairs from drifting sideways — a danger most people never think about.