Question

Eli flew home from vacation with a heavy bag. With the first airline he flew, Eli had to pay $29 to check his bag, plus $18 for every kilogram that his bag was over the weight limit. The next flight was with another airline that had the same weight limit. Eli had to pay $19 per kilogram that his bag was over the weight limit, in addition to the checked bag fee of $18. By coincidence, the fees ended up being the same with both airlines. solve answer without using C_2

Answer 1

To solve the problem, we need to set up equations based on the fees charged by both airlines.

Let's denote:
- \( w \): the weight of Eli's bag (in kilograms)
- \( l \): the weight limit (in kilograms)
- \( x \): the amount by which the bag is over the weight limit (i.e., \( x = w - l \))

The fees for the two airlines can be expressed as follows:

1. **First Airline**:
- Checked bag fee = $29
- Additional fee for being over weight = $18 per kilogram over the limit
- Total fee:
\[
\text{Fee}_1 = 29 + 18x
\]

2. **Second Airline**:
- Checked bag fee = $18
- Additional fee for being over weight = $19 per kilogram over the limit
- Total fee:
\[
\text{Fee}_2 = 18 + 19x
\]

According to the problem, the fees from both airlines are the same, so we can set the two equations equal to each other:
\[
29 + 18x = 18 + 19x
\]

Now, let's solve for \( x \).

1. Subtract \( 18x \) from both sides:
\[
29 = 18 + x
\]

2. Subtract 18 from both sides:
\[
29 - 18 = x
\]
\[
x = 11
\]

Now we know that Eli's bag is 11 kilograms over the weight limit. To find the weight of Eli's bag, we can express it in terms of the weight limit:
\[
w = l + x = l + 11
\]

Next, we can substitute \( x = 11 \) into either fee equation to find the total fee. Let's use the equation for the first airline:
\[
\text{Fee}_1 = 29 + 18(11) = 29 + 198 = 227
\]

So regardless of the weight limit, both airlines charged Eli \( \$227 \) for checking his bag.

In summary:
- The amount Eli's bag was over the weight limit: **11 kilograms**
- The total fee paid by Eli: **$227**