When Lines Never Cross

Two lines do not always cross. Parallel lines have the same slope and different intercepts, so they share no point. A system of two parallel lines has no solution. For example, $x+y = 3$ and $x+y = 7$ cannot both be true. No pair of numbers adds to 3 and to 7 at the same time. The algebra shows it too. Subtracting the first equation from the second leaves $0 = 4$, which is false.

Equivalent equations describe the same line. For example, $4x+2y = 10$ is twice $2x+y = 5$, so every point on that line satisfies both. The system has infinitely many solutions. Two distinct intersecting lines have one solution, and two distinct parallel lines have none.