To find a line that is perpendicular to the line \( y = -\frac{7}{4}x \) and goes through the point \((-7, 5)\), we need to determine the slope of the perpendicular line.
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Determine the slope of the given line: The slope of the line \( y = -\frac{7}{4}x \) is \(-\frac{7}{4}\).
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the negative reciprocal of \(-\frac{7}{4}\) is: \[ m = \frac{4}{7} \]
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Use the point-slope form of the equation of a line: The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point through which the line passes, and \( m \) is the slope.
Substituting \( m = \frac{4}{7} \) and the point \((-7, 5)\): \[ y - 5 = \frac{4}{7}(x + 7) \]
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Simplify this equation: Expanding it, we get: \[ y - 5 = \frac{4}{7}x + 4 \] \[ y = \frac{4}{7}x + 9 \]
Now, we can match this to the given options:
- A: \( y = -\frac{7}{4}x + 5 \) (not correct)
- B: \( y = \frac{4}{7}x + 9 \) (this matches)
- C: \( y = \frac{4}{7}x + 5 \) (not correct)
- D: \( y = -\frac{4}{7}x + 1 \) (not correct)
Thus, the correct answer is:
B: \( y = \frac{4}{7}x + 9 \).