To find the equation of line \(d\), we need to determine two things: the slope of line \(c\) and then the slope of line \(d\) since it is perpendicular to line \(c\).
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Identify the slope of line \(c\): The equation of line \(c\) is given as: \[ y = -6x - 3 \] From this equation, we can see that the slope (\(m_c\)) of line \(c\) is \(-6\).
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Calculate the slope of line \(d\): The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope (\(m_d\)) of line \(d\) will be: \[ m_d = -\frac{1}{m_c} = -\frac{1}{-6} = \frac{1}{6} \]
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Use the point-slope form to find the equation of line \(d\): Line \(d\) passes through the point \((3, -3)\) and has a slope of \(\frac{1}{6}\). We use the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] Plugging in the values: \[ y - (-3) = \frac{1}{6}(x - 3) \] This simplifies to: \[ y + 3 = \frac{1}{6}(x - 3) \] Multiplying both sides by 6 to eliminate the fraction: \[ 6(y + 3) = x - 3 \] Distributing the 6: \[ 6y + 18 = x - 3 \] Rearranging to get \(y\) by itself: \[ 6y = x - 3 - 18 \] \[ 6y = x - 21 \] Finally, divide everything by 6 to express \(y\) in slope-intercept form: \[ y = \frac{1}{6}x - \frac{21}{6} \] We can simplify \(-\frac{21}{6}\): \[ y = \frac{1}{6}x - \frac{7}{2} \]
Therefore, the equation of line \(d\) in slope-intercept form is: \[ \boxed{y = \frac{1}{6}x - \frac{7}{2}} \]