A line passes through the point \( (0,3) \) and is perpendicular to another line with equation \( y=2 x+1 \). What is the equation of the line?
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Answer
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Bonus Knowledge
To find the equation of the line that passes through the point \( (0,3) \) and is perpendicular to the line given by \( y = 2x + 1 \), we first need to determine the slope of the original line. The slope of the line \( y = 2x + 1 \) is \( 2 \). Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, the slope of the line we want to find will be \( -\frac{1}{2} \). Now we can use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. Plugging in the point \( (0,3) \) and the slope \( -\frac{1}{2} \): \[ y - 3 = -\frac{1}{2}(x - 0). \] Simplifying this gives: \[ y - 3 = -\frac{1}{2}x, \] or \[ y = -\frac{1}{2}x + 3. \] Thus, the equation of the line is \( y = -\frac{1}{2}x + 3 \).