Point-slope form calculator
Enter a point and a slope and get the line's equation two ways: the point (1, 2) with slope 3 gives y − 2 = 3(x − 1), and the same line crosses the y-axis at −1. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 1
- Y
- 2
- Slope
- 3
- Intercept
- -1
- Point slope
- y - 2 = 3(x - 1)
- Slope intercept
- y = 3x - 1
The point-slope form of a line is the fastest way to write an equation when you already know one point on the line and the slope. Students meet y minus y-zero equals m times open parenthesis x minus x-zero close parenthesis in algebra courses, entrance exams, and coordinate-geometry units, and automation scripts need the same identity without pulling a full computer-algebra system.
What point-slope form means and how it becomes y = mx + b
Point-slope form encodes the definition of slope directly. If a non-vertical line has slope m and passes through a fixed point with coordinates x-zero and y-zero, then for every other point on the line the rise over run equals m, which rearranges to the textbook identity y minus y-zero equals m times open parenthesis x minus x-zero close parenthesis. That single equation already describes the unique non-vertical line determined by the point and the slope.
Expanding and collecting terms yields the slope-intercept form y equals m x plus b, where the intercept b is computed as y-zero minus m times x-zero. Both writings represent the same geometric line: point-slope is ideal when the known data is a point and a gradient, while slope-intercept is convenient for graphing, for reading the y-intercept, and for comparing two lines that already share the y equals m x plus b pattern.
When the slope is zero the line is horizontal, the point-slope equation simplifies to a constant difference on y equal to zero, and the slope-intercept form collapses to y equals the constant y-zero. The calculator always echoes the rounded inputs, the numeric intercept, the point-slope string, the slope-intercept string, and a short formula reminder so clients can display the derivation next to the answer without re-deriving the algebra locally.
What to enter and what you get back
The specification that a non-numeric slope raises an error is intentional product behaviour: many classroom and form UIs collect slope as free text, and a hard failure is clearer for grading and retries than a silent default. Missing coordinates fail the same way so callers never receive a half-built equation.
Example: x equals one, y equals two, slope equals three yields point-slope y minus two equals three times open parenthesis x minus one close parenthesis and slope-intercept y equals three x minus one because b equals two minus three times one equals negative one. Failed validation is not billed.
Where a point-slope form calculator earns its keep
Algebra classrooms and tutoring platforms use a point-slope form calculator so learners can check both the point-slope writing and the rearranged slope-intercept writing for arbitrary points and slopes without sign errors. Engineering notebooks often start from a measured sensor point and a calibrated gradient—exactly the point-plus-slope data model—and need y equals m x plus b for later interpolation without importing a symbolic math stack.
Related KIT tools solve neighbouring intents: the two-point line equation cap when you know two points rather than a slope, the slope calculator when you only need m from two points, and the standard-form converter when you already have slope-intercept and need A x plus B y plus C. Keep requests small, pin the rounded intercept and equation strings in tests, and you get reproducible line forms for homework, LMS plugins, and edge workers without standing up computer-algebra infrastructure of your own.
What you can do with it
Classroom and homework checks
Let students enter a point and a slope, then verify both the point-slope equation and the slope-intercept rearrangement against textbook methods.
Sensor and calibration notebooks
Turn a measured point plus a known gradient into y = mx + b for interpolation scripts without importing a full computer-algebra library.
FAQ
How do I get the point-slope equation of a line?
Send finite numbers x and y for a known point and slope for m.
What happens if the slope is not a number?
Missing, empty, or non-numeric slope values are rejected and are not billed.
How is the y-intercept computed?
From the identity b = y0 − m·x0.
Does this handle vertical lines?
No. Vertical lines have undefined (infinite) slope. This calculator requires a finite slope; use a two-point or standard-form tool for vertical cases.
What is the difference between point-slope and slope-intercept form?
They describe the same line. Point-slope suits the moment when you know a point and a slope; slope-intercept is handier for graphing, for reading where the line crosses the y-axis and for comparing two lines.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geom/point-slope-form \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":1,"y":2,"slope":3}'const res = await fetch("https://api.kit.forhosting.com/geom/point-slope-form", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 1,
"y": 2,
"slope": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/point-slope-form",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 1,
"y": 2,
"slope": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/point-slope-form", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":1,"y":2,"slope":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":1,"y":2,"slope":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/point-slope-form", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 1,
"y": 2,
"slope": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.point_slope_form",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This point-slope form calculator accepts the coordinates of a known point and a finite slope m, returns a clean point-slope equation string, rearranges to slope-intercept form y equals m x plus b with b equal to y-zero minus m times x-zero, and fails with invalid_input when the slope is missing or non-numeric instead of inventing a vertical-line special case.
What point-slope form means and how it becomes y = mx + b
Vertical lines are intentionally out of scope because a vertical line does not have a finite slope; callers that only know a vertical direction should use a two-point or standard-form tool instead of forcing an infinite m into this endpoint.
Input contract, response shape, and non-numeric slope errors
POST a JSON body with three required finite numbers: x, the x-coordinate of the known point; y, the y-coordinate of that same point; and slope, the rise-over-run value m of the line. Each field may arrive as a JSON number or as a numeric string with an optional sign, decimal point, or scientific exponent; empty strings, free text, objects, arrays, NaN, and Infinity fail with invalid_input that names the offending field. On success the body includes the echoed x, y, and slope; intercept as the computed b rounded for stable goldens; point_slope as a human-readable string such as y minus two equals three times open parenthesis x minus one close parenthesis; slope_intercept as y equals three x minus one; and formula restating the two classical identities. Pair the response with your own plotter if you need a sketch; this endpoint solves the algebraic forms only and does not emit graphs or multi-step prose beyond the formula field.
Where a point-slope form calculator earns its keep
Exam-prep generators and LMS plugins need a deterministic oracle that rejects garbage slope strings with invalid_input instead of returning a plausible-looking wrong intercept that a buggy grader might accept.
Integration use cases
Exam and LMS auto-grading
Mark point-slope items with a deterministic oracle that rejects non-numeric slope as invalid_input and never invents a vertical-line special case.
Technical questions
What does the point-slope form calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.