Point Slope Form Calculator

What this calculator does

So, you want the equation of a line in point-slope form, y minus y1 = m(x minus x1). This tool writes it from what you know: two points, a point and the slope, or the slope and the y-intercept.

A dropdown selects which of those you are starting from, and the boxes adjust to suit.

How to use it

1. Choose your known information: two points, one point and the slope, or the y-intercept and the slope.
2. Enter the values.
3. Press Calculate.

If you give it two points, it finds the slope first, then drops one of your points into the form.

The form: y minus y1 = m(x minus x1)

Point-slope form writes a line as

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is any one point the line passes through. Unlike slope-intercept form, it does not ask for the y-intercept. It asks only for a slope and a single point, which is often exactly what you have in front of you.

Where it comes from

This form is not a new rule to memorise, it is the slope definition rearranged. Slope between a fixed point (x1, y1) and a general point (x, y) on the line is

m = (y - y1) / (x - x1)

Multiply both sides by (x - x1) to clear the fraction, and you get y - y1 = m(x - x1). That is the whole derivation. Point-slope form is just the statement that the slope from your known point to any other point on the line is always the same m.

Why it is the natural first form to write

The moment you know a slope and a point, you can write point-slope form instantly, with no working. You do not have to solve for the y-intercept the way slope-intercept form needs you to; you just slot the slope and the point straight in. That makes it the form to reach for first in a lot of problems, whether you are handed a point and a slope directly, or you have found the slope from two points and want to capture the line before tidying it into anything else.

A worked example, step by step

Take a line with slope 2 through the point (1, 3).

• Slope m = 2, point (x1, y1) = (1, 3).
• Drop them in: y - 3 = 2(x - 1).

That is the equation, written immediately from the two facts, no rearranging required.

Turning it into other forms

Point-slope form is often the starting line, not the finish. Multiply out the bracket and solve for y and it becomes slope-intercept form, y = mx + b. Gather the x and y terms on one side instead and it becomes standard form, Ax + By = C. The slope-intercept form calculator produces the first, and the equation of a line calculator lays out all three forms together, so you can see the same line expressed each way.

Questions people ask

What is point-slope form good for?

Writing a line's equation the instant you know its slope and one point on it, without first working out the y-intercept.

What can I start from?

Two points, a single point plus the slope, or the slope plus the y-intercept. The tool handles all three.

Where does the form come from?

From the slope formula. Rearranging m = (y - y1) / (x - x1) by clearing the fraction gives y - y1 = m(x - x1).

If I have two points, which one goes in?

Either one works. Both points lie on the line, so using one or the other gives equations that describe the same line.

How do I get slope-intercept or standard form?

Expand and solve for y for slope-intercept form, or move the terms to one side for standard form. The equation of a line calculator does all three at once.

References

On point-slope form. A line through (x1, y1) with slope m can be written as y minus y1 = m(x minus x1), which is the slope relation rearranged.

1. Eric W. Weisstein, "Point-Slope Form," from MathWorld, a Wolfram resource, on y minus y1 = m(x minus x1) for a line through a point with a given slope.
2. Christopher Stover and Eric W. Weisstein, "Line," from MathWorld, a Wolfram resource, on the line and its standard forms.