Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Gavin flew home from vacation with a heavy bag. With the first airline he flew, Gavin had to pay $\$18$ to check his bag, plus $\$12$ for every kilogram that his bag was over the weight limit. The next flight was with another airline that had the same weight limit. Gavin had to pay $\$11$ per kilogram that his bag was over the weight limit, in addition to the checked bag fee of $\$23$ . By coincidence, the fees ended up being the same with both airlines. How far over the weight limit was the bag? How large was each airline's fee? $\newline$ The bag was _ kilograms over the weight limit, and each airline's fee was $\$$ _.

Full solution

Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Gavin flew home from vacation with a heavy bag. With the first airline he flew, Gavin had to pay

\$18

to check his bag, plus

\$12

for every kilogram that his bag was over the weight limit. The next flight was with another airline that had the same weight limit. Gavin had to pay

\$11

per kilogram that his bag was over the weight limit, in addition to the checked bag fee of

\$23

. By coincidence, the fees ended up being the same with both airlines. How far over the weight limit was the bag? How large was each airline's fee?

\newline

The bag was ___ kilograms over the weight limit, and each airline's fee was

\$

___.

Formulate Equations: Let's denote the weight over the limit as $x$ kilograms and the fee charged by each airline as $F$ dollars. $\newline$ The first airline's fee can be represented by the equation: $\newline$ $F = 18 + 12x$ $\newline$ The second airline's fee can be represented by the equation: $\newline$ $F = 23 + 11x$ $\newline$ Since the fees ended up being the same, we can set these two equations equal to each other to form a system of equations.
Write System of Equations: Write the system of equations based on the information given: $\newline$ $18 + 12x = F$ $\newline$ $23 + 11x = F$ $\newline$ Since $F$ is the same for both equations, we can set the right sides of the equations equal to each other: $\newline$ $18 + 12x = 23 + 11x$
Solve for x: Solve for x by subtracting $11x$ from both sides of the equation: $\newline$ $18 + 12x - 11x = 23 + 11x - 11x$ $\newline$ $18 + x = 23$
Calculate Bag Weight: Subtract $18$ from both sides to solve for $x$ : $\newline$ $x = 23 - 18$ $\newline$ $x = 5$ $\newline$ This means Gavin's bag was $5$ kilograms over the weight limit.
Find Airline Fees: Now, we can find the fee charged by each airline by substituting $x$ back into either of the original equations. Let's use the first airline's equation: $\newline$ $F = 18 + 12x$ $\newline$ $F = 18 + 12(5)$ $\newline$ $F = 18 + 60$ $\newline$ $F = 78$ $\newline$ So, each airline's fee was $\$78$ .