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How many solutions does the following system of linear equations have? -x + 5y = -1 x - y = 2 x + 3y = 3 A. infinitely many B. two distinct solutions C. unique D. none

GATE 2004 · Engineering Mathematics · System of Equations · medium

Answer: The system has a unique solution: x = 9/4, y = 1/4. Answer: C. unique.

  1. Solve equations 2 and 3 simultaneously: Eq2: x - y = 2 => x = y + 2. Substitute into Eq3: (y+2) + 3y = 3 => 4y = 1 => y = 1/4. Then x = 1/4 + 2 = 9/4.
  2. Check candidate in equation 1: Eq1: -x + 5y = -1. Substitute: -(9/4) + 5(1/4) = -9/4 + 5/4 = -4/4 = -1. This equals -1, so Eq1 IS satisfied.
  3. Verify all three equations are satisfied: With x=9/4, y=1/4: Eq1: -(9/4)+5*(1/4) = -9/4+5/4 = -4/4 = -1. Satisfied. Eq2: 9/4-1/4 = 8/4 = 2. Satisfied. Eq3: 9/4+3*(1/4) = 9/4+3/4 = 12/4 = 3. Satisfied. All 3 equations are satisfied by the unique candidate.