ForHosting KIT · Math

Standard form of a line calculator

2x + 4y = 6 and x + 2y = 3 are the same line; standard form keeps only the second, with whole-number coefficients, no common factor and a positive leading term. Enter a, b and c and get that one canonical equation. Free in your browser.

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Worked example

If you enter

A: 2
B: 4
C: 6

You get

A: 1
B: 2
C: 3
Equation: x + 2y = 3

Standard form of a line writes a linear relation as Ax plus By equals C, where A, B, and C are integers, the greatest common divisor of their absolute values is one, and the leading coefficient is positive so the same geometric line always has one canonical equation. Students converting slope-intercept or messy fractional coefficients, tutors checking homework, and automation scripts that store lines for graphing or constraint systems all need that unique triple without sign flips or common factors left behind. This standard form of a line calculator accepts finite coefficients a, b, and c for the equation ax plus by equals c, clears terminating decimals and rational denominators, divides by the integer GCD, and flips the triple when needed so A is positive—or B is positive when A is zero for horizontal lines. Vertical and horizontal flags help clients branch without re-deriving geometry.

What standard form Ax + By = C means and why normalization matters

On the Cartesian plane every straight line can be written in many algebraically equivalent ways: slope-intercept y equals mx plus k, point-slope, general form ax plus by plus d equals zero, and the school-standard form Ax plus By equals C. Two equations represent the same line whenever one is a non-zero scalar multiple of the other, so 2x plus 4y equals 6 and x plus 2y equals 3 describe identical geometry.

Without a normalization rule, databases, answer keys, and hash-based caches treat those strings as different and graders mark correct work wrong for an extra factor of two. Standard form with integer coefficients, unit GCD, and a positive leading term fixes a single representative: clear denominators so every coefficient is an integer, divide A, B, and C by gcd of their absolute values, then multiply by negative one if the leading coefficient is negative.

The leading coefficient is A when A is not zero; when A is zero the line is horizontal and the leading coefficient is B, which must then be made positive. Vertical lines keep A positive and B equal to zero, giving Ax equals C. This capability applies exactly that contract to any finite triple a, b, c that defines a real line, returning the normalized integers together with a display string such as 2x plus 3y equals 6 and booleans that mark vertical or horizontal cases for UI branching.

What you can do with it

Classroom and homework checks

Students enter a, b, and c for ax+by=c and verify the unique integer standard form Ax+By=C with positive leading coefficient against textbook answers.

Exam and LMS auto-grading

Mark standard-form items with a deterministic oracle that reduces GCD and orients the leading term so equivalent student equations compare fairly.

Graphing and constraint pipelines

Normalize fractional line equations to integer Ax+By=C before storage, hashing, or linear-programming constraint assembly without a full CAS.

FAQ

How do I convert a linear equation to standard form Ax + By = C?

Send finite coefficients a, b, and c for ax+by=c.

What does positive leading term mean?

If A is not zero, A must be positive. If A is zero (horizontal line), B must be positive. The entire triple is multiplied by −1 when needed so that rule holds.

What if a and b are both zero?

Without a non-zero coefficient of x or y, ax+by=c does not define a unique line in the plane.

Does the calculator accept fractional coefficients?

Yes. Terminating decimals are scaled by powers of ten; other rationals are approximated via continued fractions, then cleared and reduced to integers.

Why divide by the greatest common divisor?

So that each line has exactly one equation. 2x + 4y = 6 and x + 2y = 3 describe the same line, but only after dividing by 2 do they look identical, which lets an answer key recognise correct work.

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

POSThttps://api.kit.forhosting.com/algebra/standard-form-line

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/algebra/standard-form-line \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"a":2,"b":4,"c":6}'

const res = await fetch("https://api.kit.forhosting.com/algebra/standard-form-line", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "a": 2,
    "b": 4,
    "c": 6
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/algebra/standard-form-line",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "a": 2,
    "b": 4,
    "c": 6
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/standard-form-line", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"a":2,"b":4,"c":6}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"a":2,"b":4,"c":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/standard-form-line", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "a": 2,
  "b": 4,
  "c": 6
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "algebra.standard_form_line",
  "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

Per request$0.002

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

Degenerate input where both a and b are zero is rejected as invalid_input because no unique line exists.

Input contract, algorithm steps, response fields, and errors

Optional null fields are never emitted. Example: a equals two, b equals four, c equals six becomes x plus 2y equals 3 with gcd two. Example: a equals negative three, b equals six, c equals nine becomes x minus 2y equals negative three after orientation and reduction. Example fractional: a equals one half, b equals one half, c equals one becomes x plus y equals two. Invalid input is not billed; successful parse calls use the published base price. POST a JSON body with three required finite numbers: a, the coefficient of x; b, the coefficient of y; and c, the right-hand side of ax plus by equals c. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, non-finite values, and magnitudes beyond one trillion fail with invalid_input naming the field. If a and b are both zero the request fails because the equation does not define a line. The response always includes the normalized integers a, b, and c; equation as a compact display string; input_a, input_b, and input_c echoing the raw inputs; gcd as the positive divisor applied after integerization; is_vertical true when B is zero and A is not; is_horizontal true when A is zero and B is not; and a short formula string documenting the three steps.

Where a standard form of a line API earns its keep

Algebra classrooms and tutoring platforms use a standard form of a line calculator so learners can check that their rearranged slope-intercept answer matches the textbook’s integer Ax plus By equals C without arguing over common factors or overall sign. Exam generators and LMS plugins need a deterministic oracle that always returns the same triple for equivalent inputs, marking both over-scaled and sign-flipped student answers against one canonical form. Failed validation does not consume balance. Graphing tools, CAD sketches, linear-programming preprocessors, and sensor-calibration notebooks often store lines as integer constraints; calling a tiny parse API avoids pulling a full computer-algebra system just to clear fractions and reduce by GCD.

Technical questions

What does the standard form of a line calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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