Standard Form to Slope Intercept Calculator
Convert a line from standard form Ax + By = C to slope-intercept form y = mx + b. Enter A, B, and C — integers or decimals — and get the converted equation with the slope, y-intercept, and x-intercept, including exact fractions when they apply.
Example: with A (coefficient of x) 2 · B (coefficient of y) 3 · C (constant) 6 → Slope-intercept form: y = -0.6667x + 2.
- Slope (m = −A/B)-0.6667 (= -2/3)
- y-intercept (b = C/B)2
- x-intercept(3, 0)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Solve Ax + By = C for y: subtract Ax, divide by B. That gives y = (−A/B)x + C/B, so m = −A/B and b = C/B. For 2x + 3y = 6: y = −(2/3)x + 2.
The conversion is two moves
Standard form Ax + By = C hides the slope; slope-intercept form y = mx + b displays it. Getting from one to the other is two algebra moves: subtract Ax from both sides, then divide everything by B. For 2x + 3y = 6: subtract to get 3y = −2x + 6, divide by 3 to get y = −(2/3)x + 2. The general result is worth memorizing — m = −A/B and b = C/B — because it turns every future conversion into arithmetic.
The negative sign in m = −A/B is where most points are lost: it appears because Ax crossed the equals sign. Note also that standard form is not unique — 2x + 3y = 6 and 4x + 6y = 12 are the same line — but they produce the same m and b, which is one reason slope-intercept form is the standard way to compare lines.
How it’s calculated
From Ax + By = C with B ≠ 0: slope m = −A/B, y-intercept b = C/B, giving y = mx + b. x-intercept = C/A when A ≠ 0. When A, B, C are all integers, m and b are also shown as reduced fractions (via GCD). If B = 0 the line is vertical (x = C/A) and has no slope-intercept form; if A = 0 it is horizontal (y = C/B). Decimals are rounded to 4 places.
A and B equal to zero simultaneously is rejected — 0 = C is either every point or no points, not a line.
Worked conversions
| Standard form | Slope m = −A/B | Intercept b = C/B | Slope-intercept form |
|---|---|---|---|
| 2x + 3y = 6 | −2/3 | 2 | y = −(2/3)x + 2 |
| x − y = 4 | 1 | −4 | y = x − 4 |
| 4x − 2y = 8 | 2 | −4 | y = 2x − 4 |
| 5x + 10y = 20 | −1/2 | 2 | y = −(1/2)x + 2 |
Computed with m = −A/B and b = C/B; each line checks by substituting the intercepts back into Ax + By = C.
Common mistakes
- Forgetting the sign flip: moving Ax across the equals sign makes the slope −A/B, not A/B.
- Dividing only the constant by B — every term gets divided, so the x-coefficient becomes −A/B too.
- Rounding a fraction slope too early: −2/3 entered as −0.67 drifts noticeably over a long graph; keep the fraction until the end.
- Trying to convert B = 0 lines — 3x = 9 is the vertical line x = 3, which no y = mx + b can represent.
Frequently asked questions
How do you convert standard form to slope-intercept form?
Solve for y: subtract Ax from both sides of Ax + By = C, then divide by B. The shortcut is m = −A/B and b = C/B. For 2x + 3y = 6 that gives y = −(2/3)x + 2.
Why is the slope negative A over B?
Because Ax has to cross the equals sign before you divide. By = −Ax + C, and dividing by B leaves the x-coefficient at −A/B. If A and B share a sign the slope is negative; opposite signs make it positive.
What if B is zero?
Then y vanished from the equation and Ax = C is the vertical line x = C/A. Vertical lines have undefined slope, so there is no slope-intercept version — that is a limitation of the form, not of the line.
Is standard form unique for a given line?
No — multiplying A, B, and C by any nonzero constant keeps the same line, so 2x + 3y = 6 and 4x + 6y = 12 match. Conventions usually ask for integer A, B, C with A ≥ 0 and no common factor.
How do I find the intercepts from standard form directly?
Set the other variable to zero: x-intercept at x = C/A, y-intercept at y = C/B. For 2x + 3y = 6 that is (3, 0) and (0, 2) — two points that make graphing immediate.