To find the equation of a line that is parallel to the given line \( y = -6x + 2 \) and passes through the point \( (-1, 2) \), we can follow these steps:
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Determine the slope of the given line: The slope-intercept form of the line is \( y = mx + b \), where \( m \) is the slope. In our case, the slope \( m \) of the given line is \( -6 \).
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Use the same slope for the parallel line: Since parallel lines have the same slope, the slope of our new line will also be \( -6 \).
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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 \( (x_1, y_1) \) are the coordinates of the given point and \( m \) is the slope. Here, our point is \( (-1, 2) \) and our slope is \( -6 \).
Plugging the values into the point-slope form: \[ y - 2 = -6(x - (-1)) \] This simplifies to: \[ y - 2 = -6(x + 1) \]
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Distribute the slope: \[ y - 2 = -6x - 6 \]
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Add 2 to both sides to solve for \( y \): \[ y = -6x - 6 + 2 \] \[ y = -6x - 4 \]
So, the equation of the line that is parallel to the given line and passes through the point \( (-1, 2) \) is: \[ \boxed{y = -6x - 4} \]
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