Let x be the rate charged per hour of the first mechanic and let y be the rate charged per hour of the second mechanic.
We know that the first mechanic worked for 10 hours and the second mechanic worked for 5 hours, the total time of work on the car can be express as:

We know that the total amount they charged is $1550, them the expression above is equal to 1550 and we have the equation:

We also know that that the sum of the rates is equal to $195, then we have the equation:

Hence, we have the system of equations:

To find the solution of the system let's solve the second equation for y:

Now we plug this in the first equation and solve the resulting one variable equation for x:

Once we know the value of x we plug in the equation we found for y:

Therefore, the rates charged of each mechanic are:
First mechanic: $115
Second mechanic: $80