Step 1 :Define the variables: Let L be the number of hours Kayla worked lifeguarding and T be the number of hours Kayla worked tutoring.
Step 2 :Form the first equation from the total money Kayla earned: \(10L + 21T = 188\).
Step 3 :Form the second equation from the number of hours Kayla worked: \(T = 4L\).
Step 4 :Solve the system of equations to find that Kayla worked 2 hours lifeguarding and 8 hours tutoring last week.
Step 5 :Final Answer: The system of equations is \(\boxed{10L + 21T = 188}\) and \(\boxed{T = 4L}\), where L is the number of hours Kayla worked lifeguarding and T is the number of hours she worked tutoring.