Intersections and the distance from a point to a line
Finding where two lines intersect, the formula for the distance from a point to a line, the distance between parallel lines, and the area of a triangle from coordinates.
Where two lines meet
A point on both lines satisfies both equations, so the intersection is the solution of the system.
One solution: the lines intersect.
No solution: the lines are parallel.
Infinitely many solutions: they're the same line.
Distance from a point to a line
The distance from a point to a line is measured along the perpendicular: it's the shortest path. You could find the perpendicular line, then the intersection, then use the distance formula. A single formula does it all at once.
Write the line with everything on one side first, then plug the point into the left side.
Area of a triangle from coordinates
Use one side as the base and the distance from the opposite vertex to that side as the height.
Practice
Quick check
What is the distance from the origin to \(6x + 8y - 20 = 0\)?