Gradient and Line Relationships

The gradient measures the direction and steepness of a non-vertical line. It compares vertical change with horizontal change.

$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\text{rise}}{\text{run}}$

Subtract the coordinates in the same order in the numerator and denominator. Reversing both differences is safe; reversing only one changes the sign.

Parallel lines have the same direction. Non-vertical parallel lines have equal gradients; distinct vertical lines are also parallel. Perpendicular lines meet at $90^\circ$. When both gradients are defined, one is the negative reciprocal of the other.

$m_1=m_2\quad\text{(parallel)}$