Question

Elena and her husband Marc both drive to work. Elena's car has a current mileage (total distance driven) of 13,000 and she drives 15,000 miles more each year. Marc's car has a current mileage of 32,000 and he drives 15,000 miles more each year. Will the mileages for the two cars ever be equal? Explain.

Answer 1

The mileages for Elena's and Marc's cars will never be equal because they both drive the same additional amount of miles each year. Starting from different initial mileages, the difference between their mileages will always remain 19,000 miles apart.

Step-by-step explanation:

The question asks whether the mileage of Elena's and Marc's cars will ever be equal, considering that Elena starts at 13,000 miles and adds 15,000 miles each year, while Marc starts at 32,000 miles and also adds 15,000 miles each year.

Understanding the Problem

To solve this, we can approach it as a problem of finding the intersection point of two linear functions representing the year-over-year mileages of the two cars. Let's denote Elena's mileage after n years as E(n) and Marc's mileage after n years as M(n).

Equations for Elena's and Marc's mileages:

1. Elena's mileage after n years: E(n) = 13,000 + 15,000n
2. Marc's mileage after n years: M(n) = 32,000 + 15,000n

Finding the Solution

To find out if their mileages will ever be the same, we set the two equations equal to each other and solve for n:

Set E(n) = M(n): 13,000 + 15,000n = 32,000 + 15,000n

Since both sides have the same coefficient for n, the terms cancel each other out, leaving: 13,000 = 32,000, which is not true.

Therefore, the mileages of the two vehicles will never be equal because the difference in their starting mileages remains the same as both cars accrue mileage at the same yearly rate.