To find the equation of the line that is perpendicular to the given line \(-x + 5y = 14\) and passes through the point \((-5, -2)\), we can follow these steps:
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Convert the given line to slope-intercept form \((y = mx + b)\): \[ -x + 5y = 14 \implies 5y = x + 14 \implies y = \frac{1}{5}x + \frac{14}{5} \] The slope \(m\) of the given line is \( \frac{1}{5} \).
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Find the slope of the perpendicular line: For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if the slope of the given line is \( \frac{1}{5} \), the slope of the perpendicular line \(m_{\text{perpendicular}}\) is: \[ m_{\text{perpendicular}} = -5 \]
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Use the point-slope form of the line: The point-slope form of a line can be written as: \[ y - y_1 = m(x - x_1) \] Using the point \((-5, -2)\) and the slope \(-5\): \[ y - (-2) = -5(x - (-5)) \implies y + 2 = -5(x + 5) \]
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Distribute and simplify: \[ y + 2 = -5x - 25 \implies y = -5x - 27 \]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point \((-5, -2)\) is: \[ \boxed{y = -5x - 27} \]