Given the system of linear equations: x + y/2 = 9 3x + y = 10 Apply the…

An iterative method for solving a linear system, such as Gauss-Seidel, converges to the exact solution from any starting guess if and only if the spectral radius of its iteration matrix is strictly less than 1. This iteration matrix depends on which equation is used to update which variable — pairing the same two equations to the two variables the other way round produces a different iteration matrix altogether, sometimes with the opposite convergence behaviour, so the pairing must be fixed before the question is well-posed. Diagonal dominance of the coefficient matrix, for a given pairing, is only a sufficient condition for convergence, not necessary — when it fails on a row, the spectral radius itself must be computed directly to settle the question.

Apply the pairing specified in the question: 3x + y = 10 updates x, and x + y/2 = 9 updates y.

1. Solve 3x + y = 10 for x: x = 10/3 − y/3.

2. Solve x + y/2 = 9 for y (multiply by 2 first: 2x + y = 18): y = 18 − 2x.

3. Write the Gauss-Seidel updates, using the newest x immediately in the y-update: x(k+1) = 10/3 − y(k)/3, then y(k+1) = 18 − 2x(k+1).

4. Substitute the first update into the second to express y(k+1) purely in terms of y(k): y(k+1) = 18 − 2(10/3 − y(k)/3) = 34/3 + (2/3)y(k).

5. Read off the iteration matrix acting on (x(k), y(k)): its eigenvalues are 0 and 2/3, so its spectral radius is ρ = 2/3.

Cross-check with the diagonal-dominance test on this same pairing: the row solved for x (3x + y = 10) is dominant (|3| > |1|), but the row solved for y (2x + y = 18) is NOT (|1| < |2|), so the standard sufficient test alone is inconclusive here. Because diagonal dominance is only sufficient and not necessary, this failure does not decide the outcome — the directly computed spectral radius (2/3) does, and it is decisive. The pairing matters, not just the system itself: using the OTHER pairing — x + y/2 = 9 to update x and 3x + y = 10 to update y — gives a different iteration matrix with spectral radius 3/2 (greater than 1), which diverges. So the specified pairing is essential to reaching the convergent iteration; a different pairing of the same two equations would not converge.

Since ρ = 2/3 < 1, the Gauss-Seidel iteration converges to the exact solution (x = −8, y = 34) for this pairing, regardless of the starting guess.