Calculate the point-slope form equation of a line from a point and the slope, or from two points, and convert it to slope-intercept and standard form.
Point Slope Form Formula
The point-slope form writes the equation of a straight line using one known point on the line and the slope of the line.
y - y1 = m(x - x1)
- y, x = the variables of the line
- y1, x1 = the coordinates of the known point on the line
- m = the slope of the line
When you only have two points and not the slope, first find the slope from the two points, then substitute one of the points into the point-slope formula.
m = (y2 - y1) / (x2 - x1)
- x1, y1 = the coordinates of the first point
- x2, y2 = the coordinates of the second point
The calculator lets you choose what you know. Pick the point and slope mode when a single point and the slope are given, and it returns the line directly. Pick the two points mode when you have two points, and it computes the slope for you before building the equation. Either mode also rearranges the result into slope-intercept form (y = mx + b) and standard form (Ax + By = C) so you can read the line in whichever form your problem asks for.
Common Slope Values and What They Mean
The slope you enter changes the direction and steepness of the line. Use the table below to check that your slope matches the line you expect.
| Slope (m) | Direction | Meaning |
|---|---|---|
| m > 0 | Rises left to right | y increases as x increases |
| m < 0 | Falls left to right | y decreases as x increases |
| m = 0 | Horizontal | y stays the same for every x |
| Undefined | Vertical | x stays the same; the two points share an x value |
The three common ways to write the same line are shown below so you can match the calculator output to the form your assignment wants.
| Form | General Equation | Best Used When |
|---|---|---|
| Point-slope | y - y1 = m(x - x1) | You know a point and the slope |
| Slope-intercept | y = mx + b | You want the y-intercept directly |
| Standard | Ax + By = C | You need integer coefficients |
Example Problems
Example 1. A line passes through the point (2, 3) with a slope of 4. Substitute the point and slope into the formula:
y - 3 = 4(x - 2)
That is the point-slope form. Expanding it gives the slope-intercept form y = 4x - 5.
Example 2. A line passes through the points (1, 2) and (3, 8). First find the slope:
m = (8 - 2) / (3 - 1) = 6 / 2 = 3
Now use the point (1, 2) in the formula:
y - 2 = 3(x - 1)
Expanding gives y = 3x - 1.
FAQ
What is point-slope form used for?
Point-slope form lets you write the equation of a line as soon as you know one point on the line and its slope. It is the quickest starting point when you do not yet know the y-intercept, and you can rearrange it into slope-intercept or standard form afterward.
How do you find the equation if you only have two points?
Calculate the slope by dividing the change in y by the change in x, m = (y2 - y1) / (x2 - x1). Then put that slope and either one of the two points into y - y1 = m(x - x1). Both points give the same line.
What happens when the two points have the same x value?
The slope is undefined because you would divide by zero. The line is vertical, and its equation is simply x = x1. A vertical line cannot be written in point-slope form.
