To find the equation of a line perpendicular to \( y = -4x - 1 \) that passes through the point \((-2, 7)\), we need to follow these steps:
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Determine the slope of the given line.
The equation \( y = -4x - 1 \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is \(-4\). -
Find the slope of the perpendicular line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of the original line is \(-4\), the slope \( m_{\text{perpendicular}} \) is: \[ m_{\text{perpendicular}} = -\frac{1}{-4} = \frac{1}{4} \] -
Use the point-slope form of the equation of a line.
The point-slope form of a line is given by the equation: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Here, we will use the point \((-2, 7)\) and the slope \(\frac{1}{4}\): -
Substitute the values into the point-slope form. \[ y - 7 = \frac{1}{4}(x - (-2)) \] Simplifying this: \[ y - 7 = \frac{1}{4}(x + 2) \]
This equation in point-slope form is: \[ y - 7 = \frac{1}{4}(x + 2) \]
This is the required equation of the line that is perpendicular to \( y = -4x - 1 \) and passes through the point \((-2, 7)\).