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Shawn and Dorian rented bikes from two different rental shops. The prices in dollars, \( y \), of renting bikes from the two different shops for \( x \) hours is shown.

Shop Shawn used: \( y = 10 + 3.5x \)

Shop Dorian used: \( y = 6x \)

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]\[ \boxed{\phantom{y}} \][/tex]

Sagot :

GinnyAnswer
To determine how much each paid for the rental when Shawn and Dorian each rented bikes for the same number of hours and paid the same price, we need to perform the following steps:

### Step 1: Set up the equations for the rental costs

The cost of renting a bike from the shop that Shawn used is given by:
[tex]\[ y = 10 + 3.5x \][/tex]

The cost of renting a bike from the shop that Dorian used is given by:
[tex]\[ y = 6x \][/tex]

### Step 2: Set the two equations equal to each other

Since Shawn and Dorian paid the same amount for the rental, we equate the two expressions for \( y \):
[tex]\[ 10 + 3.5x = 6x \][/tex]

### Step 3: Solve for \( x \)

To find the number of hours \( x \) for which both rental costs are equal, we solve the equation:
[tex]\[ 10 + 3.5x = 6x \][/tex]

Isolate \( x \) on one side of the equation:
[tex]\[ 10 = 6x - 3.5x \][/tex]
[tex]\[ 10 = 2.5x \][/tex]

Solve for \( x \):
[tex]\[ x = \frac{10}{2.5} \][/tex]
[tex]\[ x = 4 \][/tex]

So, they rented the bikes for 4 hours.

### Step 4: Calculate the amount each paid

Substitute \( x = 4 \) back into either of the original cost equations to find the total amount paid.

Using the equation for Shawn's rental cost:
[tex]\[ y = 10 + 3.5x \][/tex]
[tex]\[ y = 10 + 3.5(4) \][/tex]
[tex]\[ y = 10 + 14 \][/tex]
[tex]\[ y = 24 \][/tex]

Using the equation for Dorian's rental cost:
[tex]\[ y = 6x \][/tex]
[tex]\[ y = 6(4) \][/tex]
[tex]\[ y = 24 \][/tex]

Both calculations yield the same amount.

### Conclusion:

Each paid [tex]\( \$24 \)[/tex] for the rental.
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