the equation of line c is y = -6x - 3. line d includes the point (3, -3) and is perpendicular to line c…

Explanation:

Step1: Find the slope of line d

The slope of line c is -6. For two perpendicular lines, the product of their slopes is - 1. Let the slope of line d be $m_d$. Then $-6\times m_d=-1$, so $m_d=\frac{1}{6}$.

Step2: Use the point - slope form to find the equation of line d

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(3,-3)$ and $m = \frac{1}{6}$. Substituting these values, we get $y-(-3)=\frac{1}{6}(x - 3)$.

Step3: Convert to slope - intercept form

Simplify the point - slope equation: [ \begin{align*} y + 3&=\frac{1}{6}(x - 3)\ y+3&=\frac{1}{6}x-\frac{1}{2}\ y&=\frac{1}{6}x-\frac{1}{2}-3\ y&=\frac{1}{6}x-\frac{1}{2}-\frac{6}{2}\ y&=\frac{1}{6}x-\frac{7}{2} \end{align*} ]

Answer:

$y=\frac{1}{6}x-\frac{7}{2}$