To find out how much Shawn and Dorian each paid for their bike rentals, we need to determine the number of hours, [tex]\( x \)[/tex], for which the total costs for both rentals are the same. Given:
- The cost function for Shawn's rental is [tex]\( y = 10 + 3.5x \)[/tex].
- The cost function for Dorian's rental is [tex]\( y = 6x \)[/tex].
Since they both paid the same amount, we can set the two cost functions equal to each other and solve for [tex]\( x \)[/tex].
[tex]\[ 10 + 3.5x = 6x \][/tex]
First, let's isolate [tex]\( x \)[/tex] by moving the variable terms to one side and the constant terms to the other side:
[tex]\[ 10 = 6x - 3.5x \][/tex]
Combine the [tex]\( x \)[/tex]-terms:
[tex]\[ 10 = 2.5x \][/tex]
Next, solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( 2.5 \)[/tex]:
[tex]\[ x = \frac{10}{2.5} \][/tex]
[tex]\[ x = 4 \][/tex]
So, they both rented the bikes for 4 hours. Now we need to find out how much each paid.
For Shawn:
[tex]\[ y = 10 + 3.5x \][/tex]
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 10 + 3.5(4) \][/tex]
[tex]\[ y = 10 + 14 \][/tex]
[tex]\[ y = 24 \][/tex]
For Dorian:
[tex]\[ y = 6x \][/tex]
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 6(4) \][/tex]
[tex]\[ y = 24 \][/tex]
Therefore, each paid $24 for the bike rental.
