To find the equation of a line that is perpendicular to the line given by \( y = -34x + 7 \) and passes through the point \( (6, 1) \), we need to follow these steps:
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Find the slope of the original line: The given line \( y = -34x + 7 \) has a slope of \( -34 \).
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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 original line's slope. Therefore, the slope \( m \) of the perpendicular line is: \[ m = -\frac{1}{-34} = \frac{1}{34} \]
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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 \( (x_1, y_1) \) is the point the line passes through. In this case, \( (x_1, y_1) = (6, 1) \) and \( m = \frac{1}{34} \).
Plugging in these values, we get: \[ y - 1 = \frac{1}{34}(x - 6) \]
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Simplify to slope-intercept form: To express this in the form \( y = mx + b \): \[ y - 1 = \frac{1}{34}x - \frac{6}{34} \] \[ y - 1 = \frac{1}{34}x - \frac{3}{17} \] \[ y = \frac{1}{34}x - \frac{3}{17} + 1 \] Since \( 1 = \frac{17}{17} \), we add: \[ y = \frac{1}{34}x - \frac{3}{17} + \frac{17}{17} \] \[ y = \frac{1}{34}x + \frac{14}{17} \]
Thus, the equation of the line in slope-intercept form is: \[ y = \frac{1}{34}x + \frac{14}{17} \]
The required value that completes the equation is: \[ \frac{14}{17} \]
Therefore, we can state that the equation is: \[ y = \frac{1}{34}x + \frac{14}{17} \]