Comprehensive Calculator Guide
The slope of a line through two points is the change in y divided by the change in x: m = (y₂ − y₁) ÷ (x₂ − x₁). The line through (1, 2) and (3, 6) has slope 2 and y-intercept 0, so its equation is y = 2x. Enter both points to get the slope and intercept instantly.
Slope and the equation y = mx + b
Slope is rise per unit of run: positive lines climb left to right, negative ones fall, horizontal lines have slope 0, and vertical lines have an undefined slope because the run is zero. Once you have m, the intercept is b = y₁ − m × x₁. For (−2, 5) and (4, 2): m = −3 ÷ 6 = −0.5 and b = 4, so y = −0.5x + 4.
Parallel lines share a slope; perpendicular slopes multiply to −1.
Grades and angles
Multiply the slope by 100 for a grade: a ramp rising 0.5 m over 6 m has slope 0.0833, an 8.33% grade, the common 1:12 ratio for wheelchair ramps. For an angle take the arctangent: 0.0833 is about 4.76°, and a slope of 1 is exactly 45°.
How to Use
- 01Enter x₁ and y₁ for the first point.
- 02Enter x₂ and y₂ for the second; the order of points does not change the slope.
- 03Read the slope and y-intercept and write y = mx + b.
Formula Used
m = (y₂ − y₁) ÷ (x₂ − x₁) | b = y₁ − m × x₁ | y = mx + bPractical Examples
Example 1: through the origin
(1, 2) and (3, 6): slope 2, intercept 0, y = 2x.
Example 2: negative slope
(−2, 5) and (4, 2): slope −0.5, intercept 4, y = −0.5x + 4.
Example 3: an access ramp
Rising 0.5 m over 6 m: slope 0.0833, an 8.33% or 1:12 grade.
Important Tips
- Subtract in the same order top and bottom: second point minus first in both.
- Check by substituting the second point into your equation.
Common Mistakes to Avoid
- Dividing the run by the rise.
- Confusing an undefined slope (vertical) with zero slope (horizontal).
Frequently Asked Questions
- How do I find the slope from two points?
- (y₂ − y₁) ÷ (x₂ − x₁): (2, 3) and (5, 9) give 6 ÷ 3 = 2.
- When is the slope undefined?
- When the line is vertical, so both points share the same x.
- How do I write the line's equation?
- Find m, then b = y₁ − m × x₁, and write y = mx + b.
- How do I turn slope into a percentage or angle?
- Percentage = slope × 100; angle = arctan(slope).