Question

I really need help solving this whole problem.

Answer 1

A. 
No, you do not have enough information to determine the price of both gasoline and oil from the prices that "you" and "your friend" paid at the station. If you look closely, you'll notice that "your friend" bought half the amount of gas and oil that you did and paid half the price-- you won't get a satisfactory answer from just this information.
b.
Yes, with this you can use elimination to solve the systems, as you bought 2 quarts of oil and 10 gallons of gas for $45.50 and the receipt owner bought 2 quarts of oil and 8 gallons of gas for $38.40.
c.
Alright, so here is elimination, that is, eliminating a variable if you add two equations together. (This is a confusing answer so I would suggest you ask someone else for a better one).
So, first we make an equation out of the word problem in the beginning.
I'll treat 1 gallon of gas as "g", and one quart of oil as "o".
     45.50 = 10g + 2o
+   38.40 = 8g   + 2o
You'll note that there are 2 "o"s in each equation. So, we'll multiply the bottom equation by -1 so the "o" cancels out and we can get the value of "g".
    45.50 = 10g + 2o
+  (38.40 = 8g + 2o) * -1
This becomes:
    45.50 = 10g + 2o
+ -38.40 = -8g  - 2o

Subtract, and this becomes

7.1 = 2g
/2
g = 3.55
Alright, so now we know the value of a gallon of gas. Let's substitute this back into the original equation to find the price for the oil. 

45.50 = 10(3.55) + 2o
45.50 = 35.5 + 2o
-35.5     -35.5
10 = 2o
/2
o = 5
So we know that 1 gallon of gas is $3.55 and one quart of oil is $5.00.