Line Equation and Slope

With the two-point formula m = (y₂ − y₁)/(x₂ − x₁): y-difference over x-difference, in the same point order. Example: P(1|2) and Q(4|11) give m = (11 − 2)/(4 − 1) = 9/3 = 3; the line climbs 3 units per step to the right. Mind two traps: first, Δy belongs in the numerator (not Δx), second, you must not mix the order, otherwise the sign flips. Intuitively m is the slope triangle: 1 to the right, m upwards. For x₁ = x₂ the formula is undefined; that is the vertical line, which has no slope.

The line falls: per step to the right it drops by |m| units. So y = −2x + 7 loses 2 units per x-step. In applied contexts a negative slope describes decrease processes, e.g. a water level sinking 2 cm per hour, or a battery losing 0.5 % per minute; m then carries the unit y-unit per x-unit. Fully sorted: m > 0 rising, m = 0 horizontal (constant function), m < 0 falling; the larger |m|, the steeper. The angle of inclination is measured negative for negative m accordingly: tan θ = m, for m = −1 that is −45°.

Three routes. First, read it off: b is the y-value where the line crosses the y-axis, i.e. f(0). Second, compute: if m is known and a point (x₁|y₁) is given, b follows from substituting into y = mx + b, rearranged b = y₁ − m·x₁. Example: m = 3 and P(1|2) give b = 2 − 3·1 = −1, so y = 3x − 1. Third, work without b: the point-slope form y = m(x − x₁) + y₁ yields the line directly. Finally check with the second point if available. Do not confuse b with the root: b lies on the y-axis, the root x₀ = −b/m on the x-axis.

Lines are parallel exactly when their slopes agree: m₁ = m₂ (and b differs, otherwise they are identical). They are perpendicular when m₁·m₂ = −1; the second slope is the negative reciprocal of the first. Example: to y = 2x + 1 every line with m = 2 is parallel and every line with m = −1/2 perpendicular, e.g. y = −0.5x + 4. This rule is the tool for normals in calculus: the normal at a curve point has slope −1/f′(x₀). Special case: horizontal (m = 0) and vertical lines are also perpendicular to each other, although the product rule formally fails there because the vertical line has no slope.

The angle of inclination θ between line and x-axis satisfies tan θ = m, so θ = arctan(m). For m = 1 that is 45°, for m = 3 about 71.6°. Traffic signs state gradients in percent: 12 % means m = 0.12, i.e. 12 m of height gain per 100 m horizontal distance; the angle is arctan(0.12) ≈ 6.8°. Conversely 100 % corresponds to exactly 45°, not 90°; that is a popular trick question. Note that percentage and angle are not proportional, because the tangent does not grow linearly. In applied tasks it pays to distinguish consistently between m (ratio), angle (degrees) and percent (m·100).