Answer :
Final answer:
No row interchanges are required to solve the given system of equations using substitution or elimination methods. Row interchanges are more relevant for matrix methods like Gaussian elimination, typically used to avoid a zero pivot or to minimize numerical instability.
Explanation:
To solve the system of equations: 13x1 + 17x2 + x3 = 5, x2 + 19x3 = 1, 12x2 - x3 = 0, we do not need to perform any row interchanges if we approach the solution using elementary row operations systematically. Since the second and third equations are already solved for one variable, we can directly use substitution or elimination methods.
However, if we were to represent this system as an augmented matrix and wish to use Gaussian elimination to solve, we could interchange rows to get a leading one in the upper left position, yet that isn't strictly necessary here given the progression of the system eliminating variables efficiently.
For systems of equations that require solving through matrix methods, such as Gaussian elimination, row interchanges are used when we need to pivot to a nonzero number in the lead diagonal position. In this case, we generally interchange rows to place the largest absolute value at the pivot position to minimize numerical instability.
Given the numerical values, no row interchange seems necessary for the stated equations as there's already a clear progression from x3 to x2, and then to x1 that can be utilized for straightforward back-substitution, which is an alternate method to Gaussian elimination.
