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A family has two cars. The first car has a fuel efficiency of 15 miles per gallon of gas and the second has a fuel efficiency of 25 miles per gallon of gas. During one particular week, the two cars went a combined total of 675 miles, for a total gas consumption of 35 gallons. How many gallons were consumed by each of the two cars that week?
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2 Answers

So here you have two variables, and two sets of equations that can be created.
 
The first set deals with the number of gallons each car utilized times their miles per gallon, canceling out gallons and leaving one with the total miles. X is the number of gallons consumed by the first car, and Y is the number consumed by the second car:
 
(15 miles/gallon * x gallons) + (25 miles/gallon * y gallons)=675 miles;
15x+25y=675
 
The second equation can be created given the information that the total gallon consumption was 35 gallons, and if X and Y are gallons consumed by the respective cars, then we can create a second equation:
x gallons+y gallons=35 gallons;
x+y=35
 
Now, if we rearrange the second equation, we can express either x as a function of y:
x=35-y
 
or y as a function of x:
y=35-x
 
We then take one of these (doesn't matter which, I'll use the first one) and replace very instance of that given variable in the first equation with what it equals as a function of the other variable.
 
Hence, here, I have taken the first equation, and each time I encounter x, I replace it with 35-y (which we just saw is equal to x):
 
15(35-y)+25y=675
 
And then it's simple as solving for y:
 
525-15y+25y=675
10y=150
y=150/10
y=15 gallons
 
And since we know x=35-y, input the new y into this equation to find x:
x=35-15
x=20 gallons
 
If you want to double check, just to make sure, put the values we got back into the first equation and see if it comes out correct:
15x+25y=675
15(20)+25(15)=675
300+375=675
675=675
 
And indeed it does! Let me know if anything in here doesn't make sense and I'd be happy to explain it again. Basically, if you need to find 2 variables, you need 2 equations, and you need to change one of the equations into a format that expresses one of the variables as a function of the other, thereafter replacing that variable in the other equation.
 
I solved it in a different way and got the same results:
(x/15) +(Y/25) = 35 Being X the amount of miles run by the first car and Y the amount of miles run by the second car
The other equation will be
X+Y=675
multiply the second equation by -3 and add it to the first that has been manipulated to obtain common denominator
5X + 3Y = 2625
-3X - 3Y = 2025
2X=600
X=300 miles run by the first car and then the number of gallons will be 300/15 = 20
Y will be 375 and the number of gallons will be 375/25 = 15
Total gallons 35 and total mileage 675 miles