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To determine which car wash charges more for the basic fee and how many extras must be chosen for the total costs to be the same, follow these steps:
### Step 1: Form the equation of the total cost for Bubbles Car Wash
We have the values of [tex]$x$[/tex] (number of extras) and corresponding [tex]$y$[/tex] (total cost) given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 4 & 6 & 8 \\ \hline y & 9 & 12 & 15 & 18 \\ \hline \end{array} \][/tex]
Since the cost is linear, it can be represented by the equation [tex]\( y = mx + c \)[/tex].
### Step 2: Calculate the slope [tex]\( m \)[/tex] for Bubbles Car Wash
Using the points from the table, calculate the slope [tex]\( m \)[/tex]:
[tex]\[ (x_1, y_1) = (2, 9) \quad \text{and} \quad (x_2, y_2) = (4, 12) \][/tex]
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 9}{4 - 2} = \frac{3}{2} = 1.5 \][/tex]
### Step 3: Calculate the intercept [tex]\( c \)[/tex] for Bubbles Car Wash
Use one of the points and the slope to find the intercept [tex]\( c \)[/tex]:
[tex]\[ y_1 = mx_1 + c \implies 9 = 1.5 \cdot 2 + c \implies 9 = 3 + c \implies c = 6 \][/tex]
So, the equation for Bubbles Car Wash is:
[tex]\[ y = 1.5x + 6 \][/tex]
### Step 4: Identify the basic fee for both car washes
- Soapy Car Wash: From the equation [tex]\( y = x + 9 \)[/tex], the basic fee is [tex]$9. - Bubbles Car Wash: From the equation \( y = 1.5x + 6 \), the basic fee is $[/tex]6.
### Step 5: Determine which car wash charges more for the basic fee
Comparing the basic fees:
- Soapy Car Wash: [tex]$9 - Bubbles Car Wash: $[/tex]6
So, Soapy Car Wash charges more for the basic fee.
### Step 6: Find the number of extras for which the total costs will be the same
Set the total cost equations equal to each other:
[tex]\[ x + 9 = 1.5x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x + 9 = 1.5x + 6 \implies 9 - 6 = 1.5x - x \implies 3 = 0.5x \implies x = \frac{3}{0.5} = 6 \][/tex]
Therefore, the number of extras that must be chosen for the total costs to be the same is 6.
### Summary
- Soapy Car Wash charges more for the basic fee.
- The number of extras for which the total costs for both car washes will be the same is 6.
### Step 1: Form the equation of the total cost for Bubbles Car Wash
We have the values of [tex]$x$[/tex] (number of extras) and corresponding [tex]$y$[/tex] (total cost) given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 4 & 6 & 8 \\ \hline y & 9 & 12 & 15 & 18 \\ \hline \end{array} \][/tex]
Since the cost is linear, it can be represented by the equation [tex]\( y = mx + c \)[/tex].
### Step 2: Calculate the slope [tex]\( m \)[/tex] for Bubbles Car Wash
Using the points from the table, calculate the slope [tex]\( m \)[/tex]:
[tex]\[ (x_1, y_1) = (2, 9) \quad \text{and} \quad (x_2, y_2) = (4, 12) \][/tex]
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 9}{4 - 2} = \frac{3}{2} = 1.5 \][/tex]
### Step 3: Calculate the intercept [tex]\( c \)[/tex] for Bubbles Car Wash
Use one of the points and the slope to find the intercept [tex]\( c \)[/tex]:
[tex]\[ y_1 = mx_1 + c \implies 9 = 1.5 \cdot 2 + c \implies 9 = 3 + c \implies c = 6 \][/tex]
So, the equation for Bubbles Car Wash is:
[tex]\[ y = 1.5x + 6 \][/tex]
### Step 4: Identify the basic fee for both car washes
- Soapy Car Wash: From the equation [tex]\( y = x + 9 \)[/tex], the basic fee is [tex]$9. - Bubbles Car Wash: From the equation \( y = 1.5x + 6 \), the basic fee is $[/tex]6.
### Step 5: Determine which car wash charges more for the basic fee
Comparing the basic fees:
- Soapy Car Wash: [tex]$9 - Bubbles Car Wash: $[/tex]6
So, Soapy Car Wash charges more for the basic fee.
### Step 6: Find the number of extras for which the total costs will be the same
Set the total cost equations equal to each other:
[tex]\[ x + 9 = 1.5x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x + 9 = 1.5x + 6 \implies 9 - 6 = 1.5x - x \implies 3 = 0.5x \implies x = \frac{3}{0.5} = 6 \][/tex]
Therefore, the number of extras that must be chosen for the total costs to be the same is 6.
### Summary
- Soapy Car Wash charges more for the basic fee.
- The number of extras for which the total costs for both car washes will be the same is 6.
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