To find the equation of a line that is perpendicular to the line \( y = -34x + 7 \) and passes through the point \( (6, 1) \), we need to determine the slope of the perpendicular line.
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Identify the slope of the given line: The slope (\( m \)) of the given line \( y = -34x + 7 \) is \( -34 \).
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Determine the slope of the perpendicular line: The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line is: \[ m_{\text{perpendicular}} = -\frac{1}{-34} = \frac{1}{34} \]
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Use the point-slope form of a line: We can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point the line passes through and \( m \) is the slope. For our case, \( (x_1, y_1) = (6, 1) \) and \( m = \frac{1}{34} \). Substituting these values into the equation, we have: \[ y - 1 = \frac{1}{34}(x - 6) \]
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Simplify to the slope-intercept form (y = mx + b): Distributing the slope: \[ y - 1 = \frac{1}{34}x - \frac{6}{34} \] Simplifying \( -\frac{6}{34} \) gives \( -\frac{3}{17} \): \[ y - 1 = \frac{1}{34}x - \frac{3}{17} \] Now, add 1 (which is \( \frac{17}{17} \)) to both sides to solve for \( y \): \[ y = \frac{1}{34}x - \frac{3}{17} + 1 \] \[ y = \frac{1}{34}x - \frac{3}{17} + \frac{17}{17} \] \[ y = \frac{1}{34}x + \frac{14}{17} \]
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Final result: The equation of the line is \[ y = \frac{1}{34}x + \frac{14}{17} \]
Now, the value that completes the equation in the form \( y = \frac{1}{34}x + b \) is \( b = \frac{14}{17} \).
So, the final answer is:
The line that is perpendicular to \( y = -34x + 7 \) and passes through the point \( (6, 1) \) is \( y = \frac{1}{34}x + \frac{14}{17} \).