Stanford University Linear Algebra Problem and Matrix Model Homework

Please help me solve the problem as detailed as possible. Provide very detailed justification. Thank you!

Problem 5: 23 points. We consider Ax = b. A is not singular. Assume that we perturb matrix A and
vector b by a small amount:
A+8A, + 86
The new solution x is itself perturbed as a result
(A + 8A) (x + 8x) = b + 8b
Show that there exists a 8A and 8b such that
||8x|l2 = || 4-12 1|84|1z ||x + 8x ||2 + || 86 ||2)
with 8x + 0.