What this calculator does
Point-slope form writes the equation of a straight line as y − y1 = m(x − x1), directly from one known point on the line, (x1, y1), and its slope, m. It is often the fastest way to write a line's equation when a point and a slope are what you are given, rather than two separate points.
The point slope equation is mathematically the same line as the more familiar y = mx + c slope-intercept form; it is just written in a way that shows the point and slope used to build it, before being rearranged. This calculator gives both forms from the same inputs, so you can see how one becomes the other.
The formula
Substitute the known point and slope directly into y − y1 = m(x − x1) for point-slope form. To reach slope-intercept form, expand the brackets and solve for y, which gives a y-intercept of c = y1 − m × x1.
| Term | Meaning |
|---|---|
| (x1, y1) | A known point that lies on the line. |
| m | The slope (gradient) of the line. |
| Point-slope form | y − y1 = m(x − x1). |
| Slope-intercept form | y = mx + c, the same line rearranged to show its y-intercept directly. |
The inputs explained
| Field | What to enter |
|---|---|
| x1 | The x-coordinate of the known point on the line. |
| y1 | The y-coordinate of the known point on the line. |
| Slope (m) | The slope of the line. A positive slope rises left to right; a negative slope falls. |
When to use it
Writing a line's equation from a point and a slope
When a problem gives one point and a slope directly, rather than two points, point-slope form is the natural way to write the equation without any extra algebra first.
Converting to slope-intercept form for graphing
Most graphing tools and further algebra expect y = mx + c, so expanding the point-slope form into slope-intercept form is a common next step once the line is written down.
Checking a homework answer
Substituting the same point and slope here confirms whether a hand-worked point-slope equation, and its slope-intercept equivalent, are correct.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the equation changes with slope, at a fixed point
The point (2, 3) held fixed, with the slope varying.
| Slope (m) | Point-slope form | Slope-intercept form |
|---|---|---|
| -3 | y − 3.000 = -3.000(x − 2.000) | y = -3.000x + 9.000 |
| -1 | y − 3.000 = -1.000(x − 2.000) | y = -1.000x + 5.000 |
| 0 | y − 3.000 = 0(x − 2.000) | y = 0x + 3.000 |
| 1 | y − 3.000 = 1.000(x − 2.000) | y = 1.000x + 1.000 |
| 2 | y − 3.000 = 2.000(x − 2.000) | y = 2.000x − 1.000 |
| 4 | y − 3.000 = 4.000(x − 2.000) | y = 4.000x − 5.000 |
| 6 | y − 3.000 = 6.000(x − 2.000) | y = 6.000x − 9.000 |
How the equation changes with the point, at a fixed slope
A slope of 4 and an x-coordinate of 2 held fixed, with the y-coordinate of the point varying.
| y1 | Point-slope form | y-intercept (c) |
|---|---|---|
| -4 | y + 4.000 = 4.000(x − 2.000) | -12.000 |
| -2 | y + 2.000 = 4.000(x − 2.000) | -10.000 |
| 0 | y − 0 = 4.000(x − 2.000) | -8.000 |
| 2 | y − 2.000 = 4.000(x − 2.000) | -6.000 |
| 3 | y − 3.000 = 4.000(x − 2.000) | -5.000 |
| 6 | y − 6.000 = 4.000(x − 2.000) | -2.000 |
Questions
What is point-slope form used for?
It is the quickest way to write a line's equation when you are given one point and the slope, rather than two points. It also makes clear, just by looking at the equation, which point and slope the line was built from.
How is point-slope form different from slope-intercept form?
They describe the same line. Point-slope form, y − y1 = m(x − x1), is built directly from a point and slope; slope-intercept form, y = mx + c, is the same line rearranged to show where it crosses the y-axis instead.
What if I only have two points, not a point and a slope?
Work out the slope from the two points first, using m = (y2 − y1) ÷ (x2 − x1), then use either point as (x1, y1) here. The gradient calculator does exactly that first step from two coordinates.
Does it matter which point on the line I use?
No. Any point on the same line, with the correct slope, produces a point-slope equation that expands to the identical slope-intercept form. The point-slope expression itself will look different depending on which point is used, but the underlying line is the same.
To find the slope and full line equation starting from two coordinate points, see the gradient calculator. For a triangle or shape built from coordinates rather than a single line, see the triangle calculator.