The pair of linear equations kx + 2y = 5 and 3x + y = 1 has a unique solution…

Key idea: A pair of linear equations has a unique solution when the determinant of the coefficient matrix is nonzero.

• Compute the determinant D of the coefficients.

  Here, D = k*1 - 2*3 = k - 6.

• Unique solution condition:

  D ≠ 0 ⇒ k - 6 ≠ 0 ⇒ k ≠ 6.

• Find the solution when k ≠ 6 (Cramer's rule):

  Dx = 5*1 - 2*1 = 3, so x = 3/(k - 6).

  Dy = k*1 - 5*3 = k - 15, so y = (k - 15)/(k - 6).

• Check the special case k = 6:

  When k = 6, the equations become 6x + 2y = 5 and 3x + y = 1. Doubling the second gives 6x + 2y = 2, which contradicts 6x + 2y = 5, so the system is inconsistent and has no solution.

Conclusion: The pair of equations has a unique solution exactly when k ≠ 6.