To find the equation of the new path that is perpendicular to the given path represented by the equation \( y = -2x - 7 \), we first need to determine the slope of the new path.
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Identify the slope of the original path: The equation \( y = -2x - 7 \) is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is -2.
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Find the slope of the perpendicular path: The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, the slope of the new path will be: \[ m_{\text{new}} = -\frac{1}{m_{\text{original}}} = -\frac{1}{-2} = \frac{1}{2} \]
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Use the point of intersection: We know that the new path intersects the original path at the point (-2, -3). We can use this point and the slope we just calculated to write the equation of the new path using the point-slope form of a line, which is given by: \[ y - y_1 = m(x - x_1) \] Substituting in the point (-2, -3) and the slope \( \frac{1}{2} \): \[ y - (-3) = \frac{1}{2}(x - (-2)) \] Simplifying this gives: \[ y + 3 = \frac{1}{2}(x + 2) \]
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Distribute and rearrange to find the equation in slope-intercept form: \[ y + 3 = \frac{1}{2}x + 1 \] Subtracting 3 from both sides: \[ y = \frac{1}{2}x + 1 - 3 \] Thus, the equation becomes: \[ y = \frac{1}{2}x - 2 \]
So the equation that represents the new path is: \[ \boxed{y = \frac{1}{2}x - 2} \]
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