To find the equation of a line that passes through the point \((-3, -1)\) and is perpendicular to the line given by the equation \(y = -\frac{1}{9}x - 6\), we need to follow these steps:
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Determine the slope of the given line: The slope of the line \(y = -\frac{1}{9}x - 6\) is \(-\frac{1}{9}\).
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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 original slope. Thus, the slope \(m\) of the line we want to find is: \[ m = -\frac{1}{(-\frac{1}{9})} = 9 \]
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Use the point-slope form to write the equation of the line: The point-slope form of the equation of a line is given by: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Using the point \((-3, -1)\) and the slope \(9\): \[ y - (-1) = 9(x - (-3)) \] Simplifying this, we get: \[ y + 1 = 9(x + 3) \]
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Distribute and rearrange to the slope-intercept form: Distributing \(9\): \[ y + 1 = 9x + 27 \] Then, subtract \(1\) from both sides: \[ y = 9x + 26 \]
Thus, the equation of the line that passes through \((-3, -1)\) and is perpendicular to the line \(y = -\frac{1}{9}x - 6\) is: \[ \boxed{y = 9x + 26} \]