Answer::
[tex]y=\frac{1}{6}x-\frac{7}{2}\text{ or }x-6y=21[/tex]Explanation:
Two lines are perpendicular if the product of their slopes is -1.
First, find the slope of line c (by comparing it with the slope-intercept form).
[tex]\begin{gathered} y=mx+b \\ y=-6x-3 \\ \implies\text{Slope, m=-6} \end{gathered}[/tex]Let the slope of line d = n
Since Lines c and d are perpendicular lines, thus:
[tex]n\times-6=-1\implies n=\frac{1}{6}[/tex]So, we have that line d has the following properties:
• Slope = 1/6
,• Point = (3,-3)
We find the equation of line d using the point-slope form:
[tex]y-y_1=m(x-x_1)\text{ where }\begin{cases}m=\frac{1}{6} \\ (x_1,y_1)=(3,-3)\end{cases}[/tex]Substitute the given values:
[tex]\begin{gathered} y-(-3)=\frac{1}{6}(x-3)\text{ } \\ y+3=\frac{1}{6}x-\frac{3}{6} \\ y=\frac{1}{6}x-\frac{1}{2}-3 \\ y=\frac{1}{6}x-\frac{7}{2} \\ \text{Multiply all through by 6} \\ 6y=x-21 \\ x-6y=21 \end{gathered}[/tex]The equation of line d is:
[tex]y=\frac{1}{6}x-\frac{7}{2}\text{ or }x-6y=21[/tex]
