What is the formula for slope between two points?

The slope formula is m = (y₂ − y₁) / (x₂ − x₁), also written as rise over run or Δy/Δx. Subtract the y-coordinates to get the rise (vertical change) and subtract the x-coordinates in the same order to get the run (horizontal change), then divide. Example with (1, 2) and (4, 6): • Rise = y₂ − y₁ = 6 − 2 = 4 • Run = x₂ − x₁ = 4 − 1 = 3 • Slope = 4/3 ≈ 1.3333 Key rules: • Order must be consistent — if you use point 2 first for y, use point 2 first for x • Either point can be labelled (x₁, y₁) — the result is the same • Positive result: line rises left to right • Negative result: line falls left to right • Zero: horizontal line • Undefined (division by zero): vertical line (x₁ = x₂)

How do you find the equation of a line from two points?

Step 1: calculate slope m = (y₂−y₁)/(x₂−x₁). Step 2: use one point to find the y-intercept b: substitute the point into y = mx + b and solve for b = y₁ − m·x₁. Step 3: write the equation y = mx + b. For (1, 2) and (4, 6): • m = (6−2)/(4−1) = 4/3 • b = 2 − (4/3)(1) = 2 − 4/3 = 2/3 • Equation: y = (4/3)x + 2/3 • Check: at x=4, y = (4/3)(4) + 2/3 = 16/3 + 2/3 = 18/3 = 6 ✓ Three equivalent forms: • Slope-intercept: y = mx + b • Point-slope: y − y₁ = m(x − x₁) • Standard: ax + by = c (multiply through to clear fractions)

What does slope tell you about a line?

Slope measures how steep a line is and in which direction it goes. A slope of 2 means the line rises 2 units for every 1 unit it moves right. A slope of −0.5 means the line falls 0.5 units for every 1 unit it moves right. Slope interpretation: • m > 0: positive slope, line rises left to right (↗) • m 1: steeper than 45° • |m| = 1: exactly 45° • |m| < 1: gentler than 45° Real-world slopes: • Road grade: a 6% grade means m = 0.06 (rises 6cm per 100cm) • Roof pitch: 4/12 pitch means m = 1/3 (rises 4 inches per 12 inches) • Wheelchair ramp maximum: m = 1/12 ≈ 0.083

How do you find the distance between two points?

Use the distance formula: d = √((x₂−x₁)² + (y₂−y₁)²). This comes directly from the Pythagorean theorem — the two points form the corners of a right triangle with legs |Δx| and |Δy|, and the distance is the hypotenuse. For (1, 2) and (4, 6): • Δx = 4 − 1 = 3, Δy = 6 − 2 = 4 • d = √(3² + 4²) = √(9 + 16) = √25 = 5 Note: this is a 3-4-5 right triangle! The distance is always positive regardless of direction. The order of the points does not matter — (x₁,y₁) to (x₂,y₂) gives the same distance as (x₂,y₂) to (x₁,y₁).

How do you find the midpoint between two points?

The midpoint formula averages the x-coordinates and y-coordinates separately: M = ((x₁+x₂)/2, (y₁+y₂)/2). The result is the point exactly halfway between the two given points. For (1, 2) and (4, 6): • Midpoint x = (1 + 4)/2 = 5/2 = 2.5 • Midpoint y = (2 + 6)/2 = 8/2 = 4 • Midpoint = (2.5, 4) Verification: the midpoint lies on the line y=(4/3)x+2/3. At x=2.5: y=(4/3)(2.5)+2/3=10/3+2/3=12/3=4 ✓ The midpoint is equidistant from both endpoints: • Distance from (1,2) to (2.5,4) = √(1.5²+2²) = √(2.25+4) = √6.25 = 2.5 • Distance from (4,6) to (2.5,4) = √(1.5²+2²) = 2.5 ✓ (both equal 2.5 = 5/2)

Slope Between Two Points Calculator

The Slope Between Two Points Calculator finds the slope, line equation, angle, distance, and midpoint from any two coordinate points instantly. Enter (x₁, y₁) and (x₂, y₂) — the slope m = (y₂−y₁)/(x₂−x₁) appears immediately as a fraction (where possible) alongside the decimal. Th...

ENTER TWO POINTS

POINT 1 (x₁, y₁)

(,)

POINT 2 (x₂, y₂)

(,)

SLOPE (m = rise/run = Δy/Δx)

4/3

= 1.33333333 (decimal)

m = (y₂−y₁)/(x₂−x₁) = (6−2)/(4−1) = 4/3 = 4/3

Line Equation

y = 4/3x + 0.6667

Angle

53.1301°

Distance

Midpoint

(2.5, 4)

y-intercept

0.666667

Perpendicular m

-3/4

↗ Positive slope — rises 4 for every 3 across

QUICK EXAMPLES

COORDINATE PLANE

STEP-BY-STEP

Step 1: Δy = y₂ − y₁ = 6 − 2 = 4

Step 2: Δx = x₂ − x₁ = 4 − 1 = 3

Step 3: m = Δy/Δx = 4/3 = 4/3

Step 4: y-intercept: b = 2 − (4/3)(1) = 0.666667

Step 5: Equation: y = 4/3x + 0.6667

Step 6: Distance = √(3²+4²) = 5

Step 7: Midpoint = ((1+4)/2, (2+6)/2) = (2.5, 4)

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter the x and y coordinates for Point 1 in the first row of inputs, and for Point 2 in the second row. Negative numbers and decimals are supported.
2
The slope appears instantly in the large black result box as a fraction (e.g. 4/3) and as a decimal. The formula application m=(y₂−y₁)/(x₂−x₁) is shown with your actual numbers.
3
Below the slope, the line equation, angle, distance, midpoint, y-intercept, and perpendicular slope are all shown.
4
Check the coordinate plane — the line is drawn through both points and extended, the rise/run triangle is shown with dashed lines, and the midpoint is marked in red.
5
Use the quick example buttons to try positive, negative, horizontal, and vertical slopes and see how each case looks on the graph.

WORKED EXAMPLE

Example: Points (1,2) and (4,6). Rise=4, run=3, slope=4/3≈1.333. y-intercept: b=2−(4/3)(1)=2/3. Line equation: y=(4/3)x+2/3. Angle=arctan(4/3)=53.13°. Distance=√(9+16)=5. Midpoint=(2.5,4). Perpendicular slope=−3/4.

FORMULAS

SLOPE & LINE FORMULAS

NAME	FORMULA	DESCRIPTION
Slope formula	m = (y₂−y₁) / (x₂−x₁)	Rise divided by run between two points
y-intercept	b = y₁ − m·x₁	Substitute one point to find b
Line equation	y = mx + b	Slope-intercept form
Distance	d = √((x₂−x₁)²+(y₂−y₁)²)	Pythagorean theorem on the coordinate pair
Midpoint	M = ((x₁+x₂)/2, (y₁+y₂)/2)	Average of x and y coordinates
Angle	θ = arctan(m)	Angle with positive x-axis
Perpendicular slope	m⊥ = −1/m	Negative reciprocal — lines meet at 90°

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Last updated: August 1, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.