Question 381808
Assume |A| is not equal to 0.  I will denote by |A(n)| the determinant of the matrix with the nth column replaced by b. Then xn = |A(n)|/|A|, by Cramer's rule. Now for n not equal to i, |A(n)| = 0, because the nth column of matrix A(n) would be k times the ith column of A(n), and by property of determinants,
 |A(n)| = 0, and so xn = |A(n)|/|A| = 0 as long as n is not equal to i.  
If n = i, then |A(i)| = k|A|, and so xi = |A(i)|/|A|= k, and the solution is complete.