Example 3: Z = 3x + 9y, multiple optimal solutionsउदाहरण 3: Z = 3x + 9y, अनेक इष्टतम हलExample 3: Z = 3x + 9y, multiple optimal solutions

NCERT Fig 12.4 · Class 12 Maths · Chapter 12 · interactive diagram

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Parts and what they mean

Feasible region ABCD (bounded) (सुसंगत क्षेत्र ABCD (परिबद्ध)) — The common region of ALL the constraints (including x ≥ 0, y ≥ 0). Every point in it, or on its boundary, is a feasible solution.
Line x + 3y = 60 (रेखा x + 3y = 60) — Boundary of x + 3y ≤ 60. It is parallel to Z = 3x + 9y (= 3(x + 3y)), which is why a whole edge is optimal.
Line x + y = 10 (रेखा x + y = 10) — Boundary of x + y ≥ 10; feasible side is away from O.
Line x = y (रेखा x = y) — Boundary of x ≤ y; feasible side is above this line (towards the y-axis).
Edge CD — every point gives Z = 180 (भुजा CD — हर बिंदु पर Z = 180) — C and D both give the maximum 180, so every point of segment CD is an optimal solution (multiple optimal solutions).
Corner A(0, 10) (कोनीय बिंदु A(0, 10)) — Z = 0 + 90 = 90.
Corner B(5, 5) — minimum (कोनीय बिंदु B(5, 5) — न्यूनतम) — x + y = 10 meets x = y: x = y = 5. Z = 15 + 45 = 60, the minimum.
Corner C(15, 15) — maximum (कोनीय बिंदु C(15, 15) — अधिकतम) — x + 3y = 60 meets x = y: 4x = 60, x = y = 15. Z = 45 + 135 = 180.
Corner D(0, 20) — maximum (कोनीय बिंदु D(0, 20) — अधिकतम) — Z = 0 + 9(20) = 180, the same maximum as at C.
x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.

Original diagram drawn by Simple Learning Hub, based on NCERT Fig 12.4. Report a mistake