To find the equation of the line that passes through the point (3, 2) and is perpendicular to the line defined by the equation \( x - 4y = -1 \), we first need to determine the slope of the given line.
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Convert to slope-intercept form (y = mx + b): \[ x - 4y = -1 \] Rearranging gives: \[ -4y = -x - 1 \quad \Rightarrow \quad y = \frac{1}{4}x + \frac{1}{4} \] Here, the slope \( m \) of the line is \( \frac{1}{4} \).
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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. Given the slope of the original line is \( \frac{1}{4} \), the slope of the perpendicular line will be: \[ m_{perpendicular} = -\frac{1}{\left(\frac{1}{4}\right)} = -4 \]
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Use point-slope form to find the equation: We have the slope and a point through which the line passes, which is (3, 2). The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting in the values: \[ y - 2 = -4(x - 3) \] Expanding this gives: \[ y - 2 = -4x + 12 \quad \Rightarrow \quad y = -4x + 14 \]
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Convert to standard form: The slope-intercept form of the line is \( y = -4x + 14 \). To convert it to standard form \( Ax + By = C \): \[ 4x + y = 14 \] Here, \( A = 4 \), \( B = 1 \), and \( C = 14 \).
Final Results:
- Slope-intercept form: \( y = -4x + 14 \)
- Standard form: \( 4x + y = 14 \)
Both forms are correct and use the smallest integer coefficients.