To find the equation of a line that is perpendicular to the line given by \( y = 2x + 9 \) and passes through the point \( (4, -1) \), we can follow these steps:
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Determine the slope of the given line: The slope of the line \( y = 2x + 9 \) is \( 2 \).
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \( m \) of the perpendicular line is: \[ m = -\frac{1}{2} \]
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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 (in this case, \( (4, -1) \)) and \( m \) is the slope.
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Substituting the values into the point-slope form: \[ y - (-1) = -\frac{1}{2}(x - 4) \] Simplifying this: \[ y + 1 = -\frac{1}{2}(x - 4) \] \[ y + 1 = -\frac{1}{2}x + 2 \] \[ y = -\frac{1}{2}x + 1 \]
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Write the final equation: The equation of the line that is perpendicular to \( y = 2x + 9 \) and passes through the point \( (4, -1) \) is: \[ y = -\frac{1}{2}x + 1 \]