To find the equation of a line that is perpendicular to the line given by the equation \( y + 2 = \frac{1}{2}x \) and which passes through the point \((-4, -2)\), we need to follow these steps:
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Find the slope of the given line: The equation is in point-slope form, where the slope (\(m\)) can be extracted. The equation \( y + 2 = \frac{1}{2}x \) implies that the slope \(m = \frac{1}{2}\).
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Determine the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore: \[ m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2 \]
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Use the point-slope form to find the equation of the perpendicular line: The point-slope form of the line can be given by the formula: \[ y - y_1 = m (x - x_1) \] where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. Substituting the point \((-4, -2)\) and the slope \(-2\): \[ y - (-2) = -2(x - (-4)) \]
Simplifying this: \[ y + 2 = -2(x + 4) \] \[ y + 2 = -2x - 8 \] \[ y = -2x - 8 - 2 \] \[ y = -2x - 10 \]
Thus, the equation of the line that is perpendicular to \(y + 2 = \frac{1}{2}x\) and passes through the point \((-4, -2)\) is: \[ \boxed{y = -2x - 10} \]