How to Find the Slope Between Two Points

Slope measures how steep a line is and in which direction it tilts. Given two points, you find it by dividing the change in the vertical direction, called the rise, by the change in the horizontal direction, called the run. In symbols, slope equals the difference of the y values divided by the difference of the x values.

A positive slope rises from left to right, a negative slope falls, a slope of zero is a flat horizontal line, and a vertical line has an undefined slope because its run is zero and you cannot divide by zero. Keeping the two points in the same order for both the top and bottom of the fraction prevents a common sign error.

Once you know the slope and one point, you can write the whole line. The slope-intercept form, y equals m x plus b, uses the slope m and the y-intercept b, which is where the line crosses the vertical axis. You get b by plugging a known point and the slope into the equation and solving.

The point-slope form is often quicker as a first step because it drops your slope and one point straight in without solving for the intercept. Both forms describe the same line, so use whichever your assignment or problem asks for.

The two points that give you a slope also give you the distance between them and the point exactly halfway along. The distance comes from the Pythagorean theorem: square the horizontal change, square the vertical change, add them, and take the square root. It is always positive because it is a length.

The midpoint is simpler still: average the two x values to get its x, and average the two y values to get its y. Distance and midpoint show up in geometry, mapping, and design work, and computing all of these at once saves you from repeating the same subtractions.

The calculator runs in your browser and draws the line for you, so you can sanity check the numbers against the picture.

If the tool reports an undefined slope, your two points share the same x value, which means the line is vertical. A slope of zero means they share the same y value and the line is horizontal. Neither is an error; both are meaningful answers.

Use the graph as a quick check. A steep line should show a slope well above one or below negative one, while a gentle line sits closer to zero. If the picture and the number disagree, you probably swapped a coordinate, so re-enter the points.