The point which does not lie in the half – plane 2x + 3y -12 ≤ 0 is
A plane is a flat two-dimensional surface extending infinitely in all directions. It can be represented using a linear equation in three variables. The general form is Ax + By + Cz + D = 0, where A, B, C are the direction ratios of the plane.
Linear Programming is a mathematical technique used to optimize a linear objective function subject to linear constraints. The solution lies within the feasible region formed by these constraints. The optimal solution is found at the vertices of this region. Linear Programming is applied in business economics and resource management for decision-making.
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In order to see which point is not in the half-plane 2x + 3y – 12 ≤ 0, we put in the coordinates for each point in the inequality.
1. Point (1, 2)
2(1) + 3(2) – 12 = 2 + 6 – 12 = -4 ≤ 0. The point (1, 2) is in the half-plane.
2. Point (2, 1)
2(2) + 3(1) – 12 = 4 + 3 – 12 = -5 ≤ 0. Point (2, 1) lies in the half-plane.
3. For point (2, 3):
2(2) + 3(3) – 12 = 4 + 9 – 12 = 1 > 0. Point (2, 3) does NOT lie in the half-plane.
4. For point (-3, 2):
2(-3) + 3(2) – 12 = -6 + 6 – 12 = -12 ≤ 0. Point (-3, 2) lies in the half-plane.
Thus, the point which does not lie in the half-plane is (2, 3).
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