NUST NET Maths · Maths · Analytic geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
A linear inequality in x and y describes a half-plane: all the points on one side of a straight line. To graph it, draw the boundary line, then test one point (the origin is easiest) to see which side to keep.
Several inequalities together give the feasible region, the set of points that satisfy every constraint at once. The conditions x ≥ 0 and y ≥ 0 keep it in the first quadrant.
In linear programming you maximise or minimise an objective function z = ax + by over this region. You do not search the whole region: the best value always occurs at a corner point, so find the corners and test each one.
Counting every intersection of two boundary lines as a corner point. An intersection is a corner only if it satisfies all the constraints, so check each one before using it.
Answer: Maximum z = 26 at (4, 2)
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