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

Sidney and Riley met for dinner in downtown Springfield. Sidney did flat-rate valet parking for $11. Riley went to another valet and paid $7 up front and $4 for every hour, including the first hour. Ultimately, the friends ended up paying the same amount. How long did they stay? How much did each one pay?

For
hours of parking, Sidney and Riley both ended up paying $
.

1 answer

To solve this problem, we can define the variables for the equations.

Let's denote:

$ x $ as the number of hours Sidney and Riley stayed.
The total amount Sidney paid can be represented as $ 11 $ dollars (flat rate).
The total amount Riley paid can be expressed as $ 7 + 4x $ dollars (where $ 4x $ represents the parking cost at $4 per hour after paying the $7 up front).

Given that they both paid the same amount, we can set up the following equation:

\[ 11 = 7 + 4x \]

Now we can solve for $ x $:

Subtract $ 7 $ from both sides:

\[ 11 - 7 = 4x \]

\[ 4 = 4x \]

Divide both sides by $ 4 $:

\[ x = 1 \]

Now that we know they stayed for $ 1 $ hour, we can calculate how much each paid.

Sidney's payment: $ 11 $ dollars (flat rate).
Riley's payment:

\[ 7 + 4 \cdot 1 = 7 + 4 = 11 \text{ dollars}. \]

Both ended up paying $ 11 $ dollars.

Therefore, we can fill in the blanks:

For 1 hour of parking, Sidney and Riley both ended up paying $11.