Slope from Standard Form of a Line

Solving Ax + By = C for y gives y = (−A/B)x + C/B, so the slope is −A/B and the y-intercept is C/B — you never need to do the rearrangement in full. Worked example: 6x + 3y = 12 has slope −2 and intercept 4, matching y = −2x + 4. Notice that C plays no part in the slope at all: 6x + 3y = 12 and 6x + 3y = −100 are parallel lines, differing only in where they sit.

That observation makes standard form the quickest way to spot parallel and perpendicular pairs: two lines are parallel when their A:B ratios match, and perpendicular when the coefficients swap with one sign flipped, so 6x + 3y = 12 is perpendicular to 3x − 6y = 5. Standard form is also the shape linear programming and systems of equations want, which is why textbooks insist on integer coefficients with A positive. The trap is the minus sign — students routinely read the slope of 6x + 3y = 12 as 2 or as 6/3 rather than −2 — and the vertical case, where B = 0 leaves x = C/A with no slope to report.