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Shawn and Dorian rented bikes from two different rental shops. The prices in dollars, [tex]\( y \)[/tex], of renting bikes from the two different shops for [tex]\( x \)[/tex] hours are shown below:

- Shop Shawn used: [tex]\( y = 10 + 3.5x \)[/tex]
- Shop Dorian used: [tex]\( y = 6x \)[/tex]

If Shawn and Dorian each rented bikes for the same number of hours and each paid the same price, how much did each pay for the rental?
[tex]\(\$\)[/tex]

Sagot :

GinnyAnswer
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.
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