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Consider the following set of equations: z + 2y = 5 4y - 3z = 3 2z + y = 15 This set A. has unique solution B. has no solution C. has finite number of solutions D. has infinite number of solutions

GATE 1998 · Engineering Mathematics · System of Equations · easy

Answer: The system has no solution (inconsistent). Answer: B.

  1. Write the augmented matrix: Rewrite all equations with variables y and z in order: y + (1/2)z ... or directly place coefficients. Equations: (1) 2y + z = 5, (2) 4y - 3z = 3, (3) y + 2z = 15. Augmented matrix: [2 1 | 5; 4 -3 | 3; 1 2 | 15].
  2. Row reduce: eliminate y from rows 2 and 3: R2 = R2 - 2*R1: [4-4, -3-2, 3-10] = [0, -5, -7]. R3 = R3 - (1/2)*R1: [0, 2-1/2, 15-5/2] = [0, 3/2, 25/2]. Matrix: [2 1 | 5; 0 -5 | -7; 0 3/2 | 25/2].
  3. Eliminate z from row 3 using row 2: R3 = R3 + (3/10)*R2: [0, 3/2 + (3/10)(-5), 25/2 + (3/10)(-7)] = [0, 3/2 - 3/2, 25/2 - 21/10] = [0, 0, 125/10 - 21/10] = [0, 0, 104/10] = [0, 0, 52/5]. Since 52/5 != 0, the system is inconsistent.
  4. Confirm by back-substitution check: From eq1 and eq2: 2y+z=5 and 4y-3z=3. Multiply eq1 by 2: 4y+2z=10. Subtract eq2: 5z=7, so z=7/5, y=(5-7/5)/2=9/5. Check eq3: y+2z=9/5+14/5=23/5 != 15. Contradiction confirmed.