Let us solve the given problem step-by-step to determine how much each paid for the bike rental.
1. Setting up the equations for both shops:
The cost [tex]\( y \)[/tex] for renting a bike from Shop Shawn for [tex]\( x \)[/tex] hours is given by:
[tex]\[
y = 10 + 3.5x
\][/tex]
The cost [tex]\( y \)[/tex] for renting a bike from Shop Dorian for [tex]\( x \)[/tex] hours is given by:
[tex]\[
y = 6x
\][/tex]
2. Equating the rental costs:
Since Shawn and Dorian each paid the same amount for the rental and rented the bikes for the same number of hours, we set the two equations equal to each other:
[tex]\[
10 + 3.5x = 6x
\][/tex]
3. Solving for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we rearrange the equation:
[tex]\[
10 + 3.5x = 6x
\][/tex]
Subtracting [tex]\( 3.5x \)[/tex] from both sides, we get:
[tex]\[
10 = 6x - 3.5x
\][/tex]
Simplifying the right side:
[tex]\[
10 = 2.5x
\][/tex]
Dividing both sides by 2.5, we find:
[tex]\[
x = \frac{10}{2.5} = 4
\][/tex]
4. Calculating the cost each paid:
Now, we substitute [tex]\( x = 4 \)[/tex] back into either of the original cost equations. We'll use the equation from Shop Shawn:
[tex]\[
y = 10 + 3.5x
\][/tex]
Substituting [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 10 + 3.5 \times 4
\][/tex]
Calculating the value:
[tex]\[
y = 10 + 14 = 24
\][/tex]
Thus, both Shawn and Dorian each paid $24 for the bike rental.
