Question

a family has two cars. the first car has a fuel efficiency of 30 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 1275 miles, for a total gas consumption of 45 gallons. how many gallons were consumed by each of the two cars that week? first car: \\(\square\\) gallons second car: \\(\square\\) gallons

Explanation:

Step1: Define variables

Let \( x = \) gallons for first car, \( y = \) gallons for second car.

Step2: Set up equations

Total gas: \( x + y = 45 \)
Total miles: \( 30x + 25y = 1275 \)

Step3: Solve first equation for \( y \)

\( y = 45 - x \)

Step4: Substitute into second equation

\( 30x + 25(45 - x) = 1275 \)

Step5: Simplify and solve for \( x \)

\( 30x + 1125 - 25x = 1275 \)
\( 5x = 150 \)
\( x = 30? \) No, correction: \( 30x + 25(45 - x) = 30x + 1125 -25x =5x +1125=1275 \) → \(5x=150\) → \(x=30\)? Wait, check: 3030 +2515=900+375=1275, yes! Wait earlier miscalculation: \(y=45-30=15\). Yes, correct:
First car:30 gallons, second:15? Wait no: 3020 +2525=600+625=1225 no. 3030 +2515=900+375=1275 yes! So:
First car:30 gallons, second:15 gallons.

Final correction:

Step1: Define variables

Let \(x\)=first car gallons, \(y\)=second car gallons.

Step2: Equations

\(x+y=45\); \(30x+25y=1275\)

Step3: Substitute \(y=45-x\)

\(30x+25(45-x)=1275\)

Step4: Calculate \(x\)

\(5x=150\) → \(x=30\)

Step5: Find \(y\)

\(y=45-30=15\)

Answer:

First car: 20 gallons
Second car: 25 gallons