To determine the type of rule represented in the table and to find the missing values, we can analyze the given pairs:
Given data:
- When \( x = 30 \), \( y = 51 \)
- When \( x = 60 \), \( y = 81 \)
Let's look for a linear relationship, where \( y = mx + b \).
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Calculate the slope (m): We have two points: \( (30, 51) \) and \( (60, 81) \).
The formula for the slope \( m \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{81 - 51}{60 - 30} = \frac{30}{30} = 1. \]
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Finding b (intercept): Use one of the points, let’s use \( (30, 51) \): \[ y = mx + b \implies 51 = 1 \cdot 30 + b \implies 51 = 30 + b \implies b = 51 - 30 = 21. \] Thus, the linear equation is: \[ y = 1x + 21 \quad \text{or} \quad y = x + 21. \]
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Calculate missing values: Now, we can find the missing values for \( x = 40 \) and \( x = 50 \):
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For \( x = 40 \): \[ y = 40 + 21 = 61. \]
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For \( x = 50 \): \[ y = 50 + 21 = 71. \]
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Now we can fill in the table:
- When \( x = 30 \), \( y = 51 \)
- When \( x = 40 \), \( y = 61 \)
- When \( x = 50 \), \( y = 71 \)
- When \( x = 60 \), \( y = 81 \)
Conclusion: The rule represented in the table is \( y = x + 21 \). The missing values are:
- \( x = 40 \): \( y = 61 \)
- \( x = 50 \): \( y = 71 \)
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