To find the r-value (correlation coefficient) for the equation \( y = -\frac{3}{2}x + 6 \), we need to understand that this is a linear equation representing a straight line. The r-value, or Pearson correlation coefficient, measures the strength and direction of the linear relationship between two variables.
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Identify the variables: Here, \( y \) is dependent on \( x \), and the equation shows a perfect linear relationship.
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Understanding linear relationships: Since this is a perfect linear equation without any errors or scatter in data, it indicates a perfect correlation.
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r-value for a perfect linear correlation: For a perfectly linear relationship:
- If the line has a positive slope, the r-value is \( +1 \).
- If the line has a negative slope, the r-value is \( -1 \).
Since the slope of the line \( y = -\frac{3}{2}x + 6 \) is negative, the r-value is:
\[ r = -1 \]
Thus, the r-value for the equation \( y = -\frac{3}{2}x + 6 \) is \( -1 \). This indicates a perfect negative linear correlation.