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:
[tex]10x+5y[/tex]We know that the total amount they charged is $1550, them the expression above is equal to 1550 and we have the equation:
[tex]10x+5y=1550[/tex]We also know that that the sum of the rates is equal to $195, then we have the equation:
[tex]x+y=195[/tex]Hence, we have the system of equations:
[tex]\begin{gathered} 10x+5y=1550 \\ x+y=195 \end{gathered}[/tex]To find the solution of the system let's solve the second equation for y:
[tex]y=195-x[/tex]Now we plug this in the first equation and solve the resulting one variable equation for x:
[tex]\begin{gathered} 10x+5(195-x)=1550 \\ 10x+975-5x=1550 \\ 5x=1550-975 \\ 5x=575 \\ x=\frac{575}{5} \\ x=115 \end{gathered}[/tex]Once we know the value of x we plug in the equation we found for y:
[tex]\begin{gathered} y=195-115 \\ y=80 \end{gathered}[/tex]Therefore, the rates charged of each mechanic are:
First mechanic: $115
Second mechanic: $80