SOLUTION: Discount Rental Cars charges a daily fee plus a mileage fee for renting its cars. Barney was charged $145.00 for 3 days and 310 miles, while Mary was charged $250.00 for 5 days and

Let "d" be the charge per day and "m" be the charge per mile.


Then from the condition, you have these two equations


    3d + 310m = 145.00  dollars    (1)   (how much Barney was charged)

    5d + 600m = 250.00  dollars    (2)   (how much Mary   was charged)


At this point, the setup is just completed.

This part of the solution is rather simple.

It is quite obvious and usually does not create any difficulties.


Thus you have this system of equations, and now your task is to solve it.


There are different methods of solution : the Substitution method;  the Elimination method and the Cramer's rule (using determinants).


I will use the Elimination method. 
For It, I multiply equation (1) by 5 (both sides) and multiply equation (2) by 3.


You will get modified equivalent system


    15d + 1550m = 725.00      (3)  

    15d + 1800m = 750.00      (4)   (how much Barney was charged)


Notice the the coefficients at "d" are equal in both equation.


Next step, subtract equation (3) from equation (4).  The terms "15d" in both equations will cancel each other, 
and you will get a single equation for one unknown "m" :


    1800m - 1550m = 750.00 - 725,    or

    250m          = 25,

       m          = 25/250 = 0.1.


Thus you just found the unknown "m" : it is 0.1  (dollar per mile).


To get the second unknown, simply substitute this value m= 0.1 into either of the two original equations.

I will substitute into equation (1). Then you get


    3d + 310*0.1 = 145,

    2d + 31      = 145,

    3d           = 145 - 31 = 114,

     d           = 114/3 = 38.


Answer.  The company charges  38 dollars per day and 0.1 dollars (= 10 cents) per mile.