What Is the Slope of a Line?
The slope of a line, often written as m, measures how steep the line is — the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. A positive slope rises left to right, a negative slope falls, a zero slope is horizontal, and a vertical line has an undefined slope.
How to Use This Calculator
Enter the coordinates of two points on the line: the first point \((x_1, y_1)\) and the second point \((x_2, y_2)\). The calculator instantly returns the slope, the rise \((\Delta y)\), the run \((\Delta x)\), the y-intercept of the line, and the angle the line makes with the horizontal axis.
The Formula Explained
The slope is computed as:
$$m = \frac{\text{Y}_2 - \text{Y}_1}{\text{X}_2 - \text{X}_1}$$
The numerator \((y_2 - y_1)\) is the rise — how much the line moves up or down. The denominator \((x_2 - x_1)\) is the run — how much it moves sideways. If the run is zero (the two x-values are equal), the line is vertical and the slope is undefined.
Worked Example
Take the points \((1, 2)\) and \((4, 8)\). The rise is \(8 - 2 = 6\) and the run is \(4 - 1 = 3\). So the slope is $$m = \frac{6}{3} = 2.$$ The y-intercept is \(b = 2 - 2 \times 1 = 0\), giving the line \(y = 2x\).
FAQ
What does a slope of 0 mean? The line is perfectly horizontal — y does not change as x changes.
Why is my slope "undefined"? Both points share the same x-value, so the run is zero. Division by zero is undefined and the line is vertical.
What is the angle of incline? It is \(\arctan(m)\) converted to degrees — the angle the line makes with the positive x-axis.
