A system of three linear equations is indeterminate (has infinitely many solutions) when the coefficient matrix is singular (rank-deficient) but the augmented system remains consistent, i.e., one equation is a linear combination of the other two on both sides.
Row1 + Row2 gives 5x + (1-1)y + 3z = 4 + a, i.e., 5x + 0y + 3z = 4+a. Comparing to Row3 (5x...
Full option-by-option analysis available to enrolled students.