Objective levels move as parallel lines across the feasible region. This lesson shows why contours stay parallel for one objective and tilt when the objective direction changes. The winning point stays hidden until the next lesson.
highlighted = computed this step
First contour
For objective z=x+y, the level z=1 is the line x+y=1. Why: a contour gathers all points with the same objective value.
z=x+y=1
Parallel contours
The levels z=2 and z=3 are parallel copies farther out. Why: increasing the value slides the same line without changing its tilt.
x+y=2x+y=3
Tilted objective
For z=3x+y, the levels z=2 and z=4 tilt differently. Why: changing the objective changes the contour direction even when the feasible polygon is the same.
z=3x+y
Direction of increase
The direction of increase is the gradient (1, 1), perpendicular to the contours. The tilted objective has gradient (3, 1). Why: the gradient points in the direction a sliding contour improves.
∇z=(1,1)
Diagram note
The diagram shows contour motion without revealing the winning corner yet. Pixel positions are rounded for layout; every number shown is exact.