Cartesian equation
ax + by + c = 0 with (a, b) ≠ (0, 0).
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Lesson overview
This lesson brings together the PDF, study notes and chapter flow so you can revise The Line in the Plane without losing the thread.
Cartesian equation
ax + by + c = 0 with (a, b) ≠ (0, 0).
Slope-intercept form
y = mx + p where m is the slope and p the y-intercept.
Slope
m = (yB − yA) / (xB − xA) for two distinct points A, B on the line.
Parallel lines
Two lines with slopes m1, m2 are parallel ⟺ m1 = m2.
Perpendicular lines
Two lines with slopes m1, m2 are perpendicular ⟺ m1 · m2 = −1.
Intersection point
Solve the system of two equations. One solution ⟺ intersecting lines.
Formula
The distance from M(x₀, y₀) to line ax + by + c = 0 is |ax₀ + by₀ + c| / √(a² + b²).
Problem-solving method
Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.
Common mistakes
Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.