Let [tex]\( x \)[/tex] represent the number of hours Shawn and Dorian rented the bikes. We need to find the number of hours [tex]\( x \)[/tex] and the amount [tex]\( y \)[/tex] they both paid, given they rented for the same duration and paid the same price.
For Shawn, the rental cost equation is:
[tex]\[ y = 10 + 3.5x \][/tex]
For Dorian, the rental cost equation is:
[tex]\[ y = 6x \][/tex]
Since both Shawn and Dorian paid the same amount, set their equations equal to each other and solve for [tex]\( x \)[/tex]:
[tex]\[ 10 + 3.5x = 6x \][/tex]
To isolate [tex]\( x \)[/tex], subtract [tex]\( 3.5x \)[/tex] from both sides:
[tex]\[ 10 = 6x - 3.5x \][/tex]
[tex]\[ 10 = 2.5x \][/tex]
Now, solve for [tex]\( x \)[/tex] by dividing both sides by 2.5:
[tex]\[ x = \frac{10}{2.5} \][/tex]
[tex]\[ x = 4 \][/tex]
Now that we have determined they rented the bikes for [tex]\( 4 \)[/tex] hours, we can find the amount [tex]\( y \)[/tex] each paid.
Substitute [tex]\( x = 4 \)[/tex] back into either of the original equations. We can use Shawn's equation for verification:
[tex]\[ y = 10 + 3.5 \cdot 4 \][/tex]
[tex]\[ y = 10 + 14 \][/tex]
[tex]\[ y = 24 \][/tex]
Thus, both Shawn and Dorian paid [tex]\( \$24 \)[/tex] for renting the bikes for [tex]\( 4 \)[/tex] hours each.
Therefore, they each paid [tex]\( \$24 \)[/tex] for the rental.
