Standard Form of a Line Calculator
This standard form of a line calculator writes a line as Ax + By = C — from two points, from a slope and an intercept, or from the coefficients themselves when you want the line read back to you.
Looking for something else? In the UK and India, standard form usually means writing a number as 3.2 × 10⁵. That is a different tool: the standard form calculator for numbers. This page is about lines.
The Three Rules That Make It Standard
An equation can be perfectly true and still not be in standard form. Three conditions have to hold at once:
| 1. Both variables left | x and y on one side, the number alone on the other |
| 2. Whole numbers | A, B and C are integers with no common factor |
| 3. A not negative | If A comes out negative, multiply everything by −1 |
So −4x − 2y = −10 is a true statement about a line, and it is not standard form: divide by −2 and you get 2x + y = 5.
There is a fourth case almost nobody states. What if A is zero? The rule about A then says nothing at all, and the convention falls to B instead — otherwise a perfectly ordinary horizontal line comes out written backwards. 0x − y = −5 should be tidied to y = 5, not left as the negative of itself.
How to Write a Line in Standard Form
From slope-intercept form. Move the x term across, then clear fractions and common factors:
y = 2/3 x − 4 → −2/3 x + y = −4 → ×3: −2x + 3y = −12 → ×(−1): 2x − 3y = 12
From two points, there is a route that avoids the slope entirely:
A = y₁ − y₂ B = x₂ − x₁ C = Ax₁ + By₁
For (1, 2) and (3, 8): A = 2 − 8 = −6, B = 3 − 1 = 2, C = −6 + 4 = −2. Divide by −2 and the answer is 3x − y = 1.
That route matters more than it looks. The usual method goes through the slope, which means dividing — and dividing is exactly what fails on a vertical line. Working straight into A and B never divides, so (4, 1) and (4, 9) need no more care than any other pair: the answer is x = 4, which is 1x + 0y = 4. Slope-intercept form cannot write that line at all.
Why Your Book May Write It Differently
Two shapes are both called standard form:
Ax + By = C and Ax + By + C = 0
They describe the same line. Moving the constant to the left reverses its sign, so 3x − y = 1 and 3x − y − 1 = 0 are the same equation twice. If your answer agrees with the book except for the sign of the last number, that is almost always why, and it is not a mistake. Check which shape your course uses and stay with it. This calculator shows both, so a mismatch never has to be a mystery.
Reading a Line Back from Ax + By = C
The three numbers give everything else, with one rearrangement. Take Ax across and divide by B:
slope = −A ÷ B y-intercept = C ÷ B x-intercept = C ÷ A
For 2x − 3y = 12 that gives a slope of 2/3, a y-intercept of −4 and an x-intercept of 6.
Those last two are worth having together. Setting one variable to zero and reading off the other is the quickest way to sketch a line by hand: mark (6, 0) and (0, −4), join them, done. Standard form gives both crossings with a single division each, which is exactly why it turns up whenever a question wants a graph rather than an equation — and why it is the form of choice for solving two lines together, since adding or subtracting two equations laid out this way eliminates a variable outright.
Watch the second one: the y-intercept is C over B, not C over A. The two get swapped often enough that at least one well-known calculator site derives it correctly and then states it the wrong way round on the same page.
Two lines have no slope to give. If B = 0 the line is vertical, and it has neither a slope nor a y-intercept. If A = 0 it is horizontal and never meets the x-axis — unless C is also 0, in which case the line is the x-axis and lies along it.
How to Use the Three Tabs
Pick what you are starting from at the top; the boxes change to match.
| Two points | Gives the standard form, with both points checked against the answer |
| Slope & intercept | Converts y = mx + b, clearing fractions and fixing the sign |
| A, B, C | Goes the other way: tidies what you typed and reads the line back |
Switching tabs clears the boxes, so nothing from the last question survives into this one. Fractions like 1/2 and decimals both work — 1/2 x + 1/3 y = 1 comes back as 3x + 2y = 6, and 6x + 4y = 2 as 3x + 2y = 1.
Open the working and you get each tidying step that actually happened — the multiplier that cleared the fractions, the common factor that was divided out, the sign flip if there was one — then both conventions and everything the three numbers tell you about the line.
Standard Form of a Line FAQ
What is the standard form of a line?
How do you convert slope-intercept form to standard form?
Why does my book write it as Ax + By + C = 0?
Does A have to be positive?
How do you find the slope from Ax + By = C?
Can a vertical line be written in standard form?
Both variables left, whole numbers, A not negative — and when A is zero, the sign falls to B.