How this instrument works
Standard form packs a line's equation as Ax + By = C, with x and y sharing one side and neither coefficient singled out. Reading off the slope and y-intercept means doing what algebra allows whenever B isn't zero: isolate y. Subtract Ax from both sides to get By = C − Ax, then divide through by B, giving y = (−A⁄B)x + (C⁄B). Match that against y = mx + b and the two constants fall out directly: m = −A⁄B, b = C⁄B.
A line's standard form is never unique — multiply A, B, and C by any nonzero constant and Ax + By = C still names the same line, so a textbook might present 2x − y = 0 as 4x − 2y = 0 just as validly. This conversion inherits that indifference: because m and b are both built as ratios, whatever scaling factor was used cancels out of the top and bottom identically, so every rescaled statement of the same line converts to the identical slope and intercept.
One coefficient forces a stop: B = 0 makes the line vertical, and a vertical line has no slope and no y-intercept to report — Ax = C fixes x no matter what y does, a shape y = mx + b can never capture. Watch the sign too: dividing by B rather than multiplying by it means a positive A over a positive B produces a NEGATIVE slope, the reverse of what two positive numbers usually suggest.
- Enter the line's three standard-form coefficients into A, in Ax + By = C, B, in Ax + By = C, and C, in Ax + By = C.
- Read Slope, m for the line's steepness, computed as −A ⁄ B.
- Read y-intercept, b for where the line crosses the y-axis, computed as C ⁄ B.
- Combine the two into y = mx + b for the finished slope-intercept equation.
- If B reads 0, the line is vertical and has no slope-intercept form — recheck the coefficients rather than trust the output.
Worked example — converting 2x − y = 0
Take the standard-form line 2x − y = 0, so A = 2, B = −1, and C = 0. The slope is m = −A ⁄ B = −2 ⁄ −1 = 2 — the two negatives cancel, leaving a positive slope of 2. The y-intercept is b = C ⁄ B = 0 ⁄ −1 = 0, since zero divided by any nonzero number stays zero.
Slope-intercept form is therefore y = 2x + 0, or simply y = 2x — a line through the origin climbing two units of y for every one unit of x. Check it against the original: at x = 1, y = 2(1) = 2, and indeed 2(1) − 2 = 0 satisfies 2x − y = 0 exactly, confirming the rearrangement lost nothing.
Questions
How do you convert standard form to slope-intercept form?
Isolate y algebraically: starting from Ax + By = C, subtract Ax from both sides to get By = C − Ax, then divide by B to reach y = (−A⁄B)x + (C⁄B). Matching that against y = mx + b gives m = −A⁄B and b = C⁄B directly, which is exactly what this sheet computes from your three coefficients.
Why is there a minus sign in the slope formula m = −A⁄B?
It comes from moving Ax to the other side before dividing: By = C − Ax rearranges to y = (−A⁄B)x + (C⁄B), so the slope always carries that negative sign. Dropping it and computing A⁄B instead is a common mistake — for A = 2, B = −1 the error would still give 2 by coincidence, but for A = 3, B = 1 it would wrongly give 3 instead of the correct −3.
What happens when B equals zero?
The line is vertical, and a vertical line like Ax = C — x fixed at C⁄A for every y — has no slope and no y-intercept. Dividing by B = 0 is undefined, and y = mx + b can never describe a vertical line no matter what values m and b take, so this sheet flags that case instead of returning a broken result.
How is this different from the standard-form calculator elsewhere on this site?
That sheet runs the opposite direction: it starts from two points and builds Ax + By = C. This one starts from an existing Ax + By = C equation and solves back out to slope m and y-intercept b. Feed that sheet's A, B, and C output into this converter and you recover the original slope and y-intercept exactly, a round trip that loses nothing.
Does it matter which scaled version of Ax + By = C I start from?
No — 2x − y = 0 and 4x − 2y = 0 describe the identical line, and both convert to the same m = 2 and b = 0, because m and b are ratios, −A⁄B and C⁄B, and any common scaling factor applied to A, B, and C cancels out of both ratios identically.
Can the y-intercept come out negative or zero?
Yes, either is normal — b = C⁄B follows whatever sign C and B carry. When C = 0, as in 2x − y = 0, b is exactly zero and the line passes through the origin; when C and B have opposite signs, b is negative and the line crosses the y-axis below it.