Slope Calculator: Slope, Intercept and Equation

Slope is rise over run — the change in y divided by the change in x between the two points. Once you have the slope, the rest of the line follows.

• Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
• Y-intercept: b = y₁ − m·x₁
• Line equation: y = mx + b
• Angle from horizontal: θ = arctan(m), in degrees
• Distance: d = √((x₂−x₁)² + (y₂−y₁)²)

Sources: standard coordinate geometry — slope formula m = (y₂−y₁)/(x₂−x₁); angle via arctangent; distance via the Pythagorean theorem.

Take the points (1, 2) and (4, 8). Rise = 8 − 2 = 6, run = 4 − 1 = 3, so the slope m = 6 ÷ 3 = 2. The intercept b = 2 − 2×1 = 0, giving the equation y = 2x. The angle is arctan(2) ≈ 63.43°, and the distance is √(3² + 6²) = √45 ≈ 6.708.

Enter the x and y coordinates of both points, then press Calculate. The calculator handles the special cases too: identical points are rejected, and vertical lines get a clear "undefined" answer instead of a broken formula.

Step by step

1. Enter point 1 (x1, y1). Either point can go first — swapping the points gives the same slope.
2. Enter point 2 (x2, y2). Decimals and negative numbers are fine.
3. Press Calculate. Read the slope, intercept, equation, angle and distance.
4. Check the note. The result explains whether the line rises, falls or is horizontal.

Slope as a fraction

A slope of 2 is the same as the fraction 2/1: two units up for every one unit across. Thinking of slope as rise-over-run keeps the geometry visible. A fraction calculator can reduce rise/run fractions like 6/3 to their simplest form when you want the exact fractional slope.

The sign and size of the slope describe the line at a glance. Positive slopes climb left to right, negative slopes descend, zero is flat, and a vertical line has no slope at all.

Slope values and their meaning

Steepness versus direction

A common mix-up is reading −3 as "less steep" than 2. Steepness comes from the absolute value: |−3| = 3 beats |2| = 2, so the −3 line is steeper even though it falls. The sign only tells you the direction.

The equation y = mx + b packs the whole line into two numbers: m sets the tilt and b sets where it crosses the y-axis. With the equation you can find y for any x without redrawing the line.

From slope and intercept to equation

Using the equation

To find the y value at x = 10 on the line y = 2x, substitute: y = 2 × 10 = 20. The intercept b is the value at x = 0, which is why it is called the y-intercept — it is where the line meets the vertical axis.

Angles come from the arctangent of the slope, and checking arctan by hand is tedious, so a scientific calculator with an inverse-tangent key is the practical way to verify the angle the calculator reports.

A vertical line like x = 3 has points (3, 1) and (3, 5): the run is 3 − 3 = 0, so rise ÷ run divides by zero and the slope is undefined. It is not zero — zero slope means a horizontal line.

Angle facts

Why the angle matters

Angles turn abstract slopes into something you can picture: a ramp, a roof pitch, a hillside. An angle near 0° is nearly flat, 45° is the diagonal you know from graph paper, and anything past 60° looks genuinely steep.