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QUESTION 11 Instructions: - Identify the independent and dependent variables for the given linear situation. - Write an equation that represents the linear relationship between the variables. - Use the equation to answer the question. Be prepared to show how you used the function equation to answer the question. Situation: There is a linear relationship between the total cost of a gym membership for the month and the number of fitness class taken that month. One month Mari took 12 classes and it cost her \( \$ 96 \). Another month she took 7 classes and paid \( \$ 56 \). Question: What is the total cost of a gym membership for the month if Mari attended 9 classes? Round final answer to the nearest cent if rounding is necessary. Answer: The total cost is \( \$ \)

Ask by Rodriquez Pearson. in the United States
Mar 13,2025

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Tutor-Verified Answer

Answer

The total cost is \$72.00.

Solution

To solve the problem, we will follow the steps outlined in the instructions. ### Step 1: Identify the Independent and Dependent Variables - **Independent Variable (x)**: The number of fitness classes taken in a month. - **Dependent Variable (y)**: The total cost of the gym membership for the month. ### Step 2: Write an Equation Representing the Linear Relationship We have two points from the situation: 1. When \( x = 12 \), \( y = 96 \) (12 classes cost $96). 2. When \( x = 7 \), \( y = 56 \) (7 classes cost $56). We can use these points to find the slope \( m \) of the linear equation \( y = mx + b \). The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points: \[ m = \frac{56 - 96}{7 - 12} = \frac{-40}{-5} = 8 \] Now that we have the slope \( m = 8 \), we can use one of the points to find the y-intercept \( b \). We'll use the point \( (12, 96) \): \[ y = mx + b \implies 96 = 8(12) + b \] \[ 96 = 96 + b \implies b = 0 \] Thus, the equation representing the linear relationship is: \[ y = 8x \] ### Step 3: Use the Equation to Answer the Question Now we need to find the total cost \( y \) when Mari attended 9 classes (\( x = 9 \)): \[ y = 8(9) \] Calculating this gives: \[ y = 72 \] ### Final Answer The total cost of a gym membership for the month if Mari attended 9 classes is: \[ \text{The total cost is } \$72.00 \]

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To determine the independent and dependent variables, we can define the number of fitness classes taken in a month as the independent variable (let's call it \( x \)), and the total cost of the gym membership as the dependent variable (let's call it \( y \)). To find the equation representing the linear relationship, we first identify two points based on the data: (12, 96) and (7, 56). We can calculate the slope \( m \) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \): \[ m = \frac{56 - 96}{7 - 12} = \frac{-40}{-5} = 8 \] Now that we have the slope, we can use one of the points to find the slope-intercept form of the equation \( y = mx + b \). Using the point (12, 96): \[ 96 = 8(12) + b \implies 96 = 96 + b \implies b = 0 \] Thus, the equation is \[ y = 8x \] Now, to find the cost when Mari attended 9 classes: \[ y = 8(9) = 72 \] So, if Mari attended 9 classes, the total cost is \( \$72.00 \).

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