Point-slope: build a line from one point
Suppose you know one point on a line, (x1, y1), and its slope m. The point-slope form writes the line directly: y − y1 = m(x − x1). It comes straight from the slope formula — slope equals rise over run between (x, y) and the known point — cleared of its denominator.
Line through (4, 1) with slope m = 2: y - y1 = m(x - x1) y - 1 = 2(x - 4) point-slope form y - 1 = 2x - 8 distribute y = 2x - 7 slope-intercept form Check (4,1): 2(4) - 7 = 8 - 7 = 1 correct
Standard form: Ax + By = C
The standard form of a line is Ax + By = C, with A, B, C usually integers and A non-negative. It treats x and y even-handedly, which makes it tidy for systems of equations and for reading off both intercepts quickly. Any line can be rewritten into it.
Convert y = (2/3)x - 4 to standard form: y = (2/3)x - 4 3y = 2x - 12 multiply through by 3 -2x + 3y = -12 move the x term left 2x - 3y = 12 multiply by -1 so A >= 0 Standard form: 2x - 3y = 12 (A=2, B=-3, C=12)
Parallel and perpendicular by slope
Parallel lines never meet, and the reason is simple: they have equal slopes. Perpendicular lines cross at a right angle, and their slopes are negative reciprocals of each other — flip the fraction and change the sign. In symbols, if one slope is m, the perpendicular slope is its negative multiplicative inverse, −1/m, and the product of the two slopes is −1.
Given line: y = (3/4)x + 1 (slope 3/4) Parallel line through (8, 0): same slope 3/4 y - 0 = (3/4)(x - 8) y = (3/4)x - 6 Perpendicular line through (8, 0): slope = negative reciprocal of 3/4 = -4/3 y - 0 = (-4/3)(x - 8) y = (-4/3)x + 32/3 Check perpendicular: (3/4)(-4/3) = -1 right angle