Step 1 :Let's denote the number of hours Mike works at job A as \(x\) and the number of hours he works at job B as \(y\).
Step 2 :From the problem, we have two equations: \(x + y = 60\) and \(6x + 7y = 381\).
Step 3 :We can write this system of equations as an augmented matrix: \[\begin{bmatrix} 1 & 1 & 60 \\ 6 & 7 & 381 \end{bmatrix}\]
Step 4 :To solve this system, we can use the method of elimination. First, we will multiply the first row by 6 and the second row by 1 to make the coefficients of \(x\) in the two equations the same: \[\begin{bmatrix} 6 & 6 & 360 \\ 6 & 7 & 381 \end{bmatrix}\]
Step 5 :Then, we subtract the first row from the second row: \[\begin{bmatrix} 6 & 6 & 360 \\ 0 & 1 & 21 \end{bmatrix}\]
Step 6 :This gives us \(y = 21\).
Step 7 :Substitute \(y = 21\) into the first equation of the original system, we get: \(x + 21 = 60\)
Step 8 :Solving for \(x\), we get \(x = 60 - 21\)
Step 9 :So, \(x = 39\)
Step 10 :Therefore, Mike works \(\boxed{39}\) hours at job A and \(\boxed{21}\) hours at job B.