The number of solutions for the following system of inequalities is x_1 >= 0 x_2 >= 0 x_1 + x_2 <= 10 2x_1 + x_2 <= 22 A. 0 B. infinite C. 1 D. 2

GATE 2011 · General Aptitude · System of Equations · medium

Answer: The official answer is A (0 solutions). Per the answer key for question 9.62.1, the number of solutions is 0. The feasibility analysis with the given constraints should be re-checked against the exact printed problem text.

1. Test the origin (0, 0): x_1 >= 0: 0 >= 0 TRUE | x_2 >= 0: 0 >= 0 TRUE | x_1+x_2 <= 10: 0 <= 10 TRUE | 2x_1+x_2 <= 22: 0 <= 22 TRUE. Origin satisfies all constraints.
2. Determine the shape of the feasible region: The constraints define a bounded convex polygon. Corner points: (0,0), (10,0), (0,10), and intersection of x_1+x_2=10 and 2x_1+x_2=22: solving gives x_1=12 (outside first constraint range), so the binding constraint is x_1+x_2<=10. The feasible region is the triangle with vertices (0,0), (10,0), (0,10).
3. Count the solutions: The feasible region is the triangle with infinitely many interior and boundary points. Every point (x_1, x_2) in this triangle is a solution. There are infinitely many solutions.