My car gets 19 miles per gallon in the city and 34 miles per gallon on the highway. Suppose I used 12.857 gallons of gas and got an average of 27 miles per gallon for the trip.

How many gallons did I use for each part of the trip?

_____gallons at 19 miles per gallon (if needed, round to 3 decimal places)

____ gallons at 34 miles per gallon (if needed, round to 3 decimal places)

put two unknowns x and y :

x is the number of liters of gasoline to move in the city
y is the number of liters of gasoline to move on the highway
x + y = 12.857 gallons
19x + 34y = 12,857.27 = 347.139
| 19x + 34y = 347.139
| x + y = 12.857
x = 6 gallon
y = 6.857 a gallon
gallons at 19 miles: x= 6 gallon
____ gallons at 34 miles: y = 6,857 gallon

thank you! but how did you get 347.139?

Total gas used for the trip = 12.857 gallons

rate for the whole trip = 27 mpg
so distance of trip = 27(12.857) miles = 347.139 miles

let the number of gallons used at the 19 mpg rate be x
then the number of gallons used at the 34 mpg rate be 12.857 - x

19x + 34(12.857 - x) = 347.139
19x + 437.138 - 34x = 347.139
-15x = -89.999
x = 6

so you used 6 gallons going at a rate of 19 mpg
and 6.857 gallons going at the 34 mpg rate

To determine how many gallons were used for each part of the trip, we can set up a system of equations using the given information.

Let's assume x represents the number of gallons used for the city portion of the trip, and y represents the number of gallons used for the highway portion of the trip.

Based on the given information:

For the city portion:
Miles per gallon in the city = 19 miles/gallon
Gallons used for the city portion = x gallons

So, the distance covered in the city portion = 19x miles.

For the highway portion:
Miles per gallon on the highway = 34 miles/gallon
Gallons used for the highway portion = y gallons

So, the distance covered in the highway portion = 34y miles.

Given that for the entire trip:
Total distance covered = 27 miles/gallon
Total gallons used = 12.857 gallons

From this, we can set up the following equations:

19x + 34y = total distance covered (27 miles/gallon * total gallons used)
x + y = total gallons used

Substituting the given values:
19x + 34y = 27 * 12.857
x + y = 12.857

Now, we can solve this system of equations to find the values of x and y.

Using either substitution or elimination method, let's solve for y in terms of x from the second equation:

y = 12.857 - x

Substituting this value of y into the first equation:

19x + 34(12.857 - x) = 27 * 12.857

Simplifying:

19x + 437.918 - 34x = 347.439
-15x = -90.479
x = (-90.479) / (-15)
x = 6.032

Therefore, the number of gallons used for the city portion is approximately 6.032 gallons (rounded to 3 decimal places).

Substituting this value back into the second equation to find y:

6.032 + y = 12.857
y = 12.857 - 6.032
y = 6.825

Therefore, the number of gallons used for the highway portion is approximately 6.825 gallons (rounded to 3 decimal places).

In summary:
Approximately 6.032 gallons were used at 19 miles per gallon.
Approximately 6.825 gallons were used at 34 miles per gallon.