Step 1 :Define the symbols L, E, and total_shirts to represent the number of hours Lara works, the number of hours Emma works, and the total number of shirts, respectively.
Step 2 :Write the equation for Lara's work as \(3L + 17 = \text{total_shirts}\). This represents the fact that Lara can embroider 3 shirts per hour and has already completed 17 shirts.
Step 3 :Write the equation for Emma's work as \(4E + 8 = \text{total_shirts}\). This represents the fact that Emma can embroider 4 shirts per hour and has already completed 8 shirts.
Step 4 :Solve the first equation for L to get \(L = \frac{\text{total_shirts}}{3} - \frac{17}{3}\).
Step 5 :Substitute the expression for L from the previous step into the second equation to get \(4E + 8 = 3(\frac{\text{total_shirts}}{3} - \frac{17}{3}) + 17\).
Step 6 :Solve this equation for E to get \(E = \frac{\text{total_shirts}}{4} - 2\).
Step 7 :Substitute the value of E into the first equation to find the total number of shirts, which gives \(\text{total_shirts} = 3L + 17\).
Step 8 :Calculate the time it will take for each tailor to finish the same total number of shirts by dividing the total number of shirts by 3, which gives \(\text{time} = L + \frac{17}{3}\).
Step 9 :Final Answer: Each tailor will have finished a total of \(\boxed{\text{total_shirts_value}}\) shirts in \(\boxed{\text{time}}\) hours.