Slope Formula

The slope formula returns a number that describes the slant of the straight line through two given points (x₁, y₁) and (x₂, y₂). The result is usually written as m.

The slope tells you two things at once. The sign of m says which way the line tilts: positive rises, negative falls, zero is flat. The size of m says how steep it is: a slope of 5 means the line climbs 5 units of y for every 1 unit of x; a slope of 0.5 means it climbs only half as much.

Use the slope formula whenever you have two points and want to know how the line through them behaves. Common situations:

• School algebra and coordinate-geometry problems.
• Drawing or analyzing a line on a graph.
• Finding the rate of change between two values (slope is the discrete version of a derivative).
• Engineering, physics, and economics: anywhere two values vary together and you want to describe the trend.
• Programming and graphics: line drawing, scrolling speed, physics simulations.

The formula is just a ratio: the change in y over the change in x. Both changes are signed, which is what makes slope able to tell direction as well as steepness.

The formula reads like a sentence: the slope is the rise divided by the run. When the run is zero (the two points share an x-coordinate), the formula is undefined and the line is vertical.

The coordinates label the two points:

• x₁ and y₁ are the coordinates of the first point.
• x₂ and y₂ are the coordinates of the second point.
• (x₁, y₁) reads as “x sub 1, y sub 1.” The subscript just labels which point this is.

Order does not matter for the answer, as long as you are consistent on both axes. If you flip which point is point 1 and which is point 2, both the rise and the run flip signs, and the slope (a ratio of those two) comes out the same.

What matters is matching the order between rise and run. If you compute rise as y₂ − y₁ then you have to compute run as x₂ − x₁, not x₁ − x₂. Mixing the order would flip the sign of one but not the other.

Take the points (1, 2) and (3, 6).

1. Identify the coordinates: x₁ = 1, y₁ = 2, x₂ = 3, y₂ = 6.
2. Compute the rise: y₂ − y₁ = 6 − 2 = 4.
3. Compute the run: x₂ − x₁ = 3 − 1 = 2.
4. Divide rise by run: m = 4 / 2 = 2.

The slope is 2. The line climbs 2 units of y for every 1 unit of x as you move from left to right.

Take the points (0, 0) and (4, 8).

• rise = 8 − 0 = 8
• run = 4 − 0 = 4
• m = 8 / 4 = 2

The slope is 2 (positive). The line rises from left to right.

Take the points (1, 5) and (3, 1).

• rise = 1 − 5 = −4
• run = 3 − 1 = 2
• m = −4 / 2 = −2

The slope is −2 (negative). The line falls from left to right.

Take the points (1, 5) and (4, 5).

• rise = 5 − 5 = 0
• run = 4 − 1 = 3
• m = 0 / 3 = 0

The slope is 0 (horizontal). The two points share a y-coordinate, so the line is perfectly flat.

Take the points (3, 1) and (3, 5).

• rise = 5 − 1 = 4
• run = 3 − 3 = 0
• m = 4 / 0 = undefined

The slope is undefined. The two points share an x-coordinate, so the run is zero, and the formula tries to divide by zero. The line is vertical. Vertical lines are sometimes described as having infinite slope, but in standard practice they simply have no defined slope.

The slope, distance, and midpoint formulas all take the same two input points but answer different questions. They are the three main coordinate-geometry calculations on a single line segment, and learning them as a set is useful because they show up together so often.

• Slope tells you how steep the line is and which way it tilts. A single number (possibly negative or undefined). m = (y₂ − y₁) / (x₂ − x₁).
• Distance tells you how long the segment is. A single non-negative number. d = √((x₂ − x₁)² + (y₂ − y₁)²). See the distance formula calculator and the Distance Formula guide.
• Midpoint tells you the center point of the segment. An ordered pair. M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2). See the midpoint formula calculator and the Midpoint Formula guide.

Same inputs, three different outputs. If you only need one of them, use the matching formula. If you are studying coordinate geometry, knowing all three together gives a full picture of any line segment.

A few traps that catch people:

• Mixing the order between rise and run. If you compute y₂ − y₁ for the rise, you must use x₂ − x₁ for the run. Flipping one but not the other produces the wrong sign.
• Putting x in the numerator. Slope is rise over run, not run over rise. The y difference goes on top.
• Treating zero rise the same as zero run. Zero rise gives a slope of 0 (horizontal line). Zero run gives an undefined slope (vertical line). They are opposite cases.
• Sign errors with negative coordinates. 5 − (−3) = 8, not 2. Subtracting a negative is adding.
• Calling a steep line a high slope without noting the sign. A slope of −5 is just as steep as a slope of 5; the line just falls instead of rising.

• Formula: m = (y₂ − y₁) / (x₂ − x₁)
• Rise is the change in y; run is the change in x. Slope is rise divided by run.
• Positive slope: line rises from left to right.
• Negative slope: line falls from left to right.
• Zero slope: horizontal line (no rise).
• Undefined slope: vertical line (no run, division by zero).
• Slope, distance, and midpoint are the three main coordinate-geometry calculations on the same two points.
• The slope calculator runs the formula in one step and labels the result by type.