First we can find the slope. The standard form of the equation of a line is:
[tex]y=ax+b[/tex]Where a is the slope and b is the intercept.
When 2 lines are perpendicular, the slopes are reciprocal and opposite to each other. If we write the given equation of the perpendicular line in the standard form we have:
[tex]6x+5y=30\rightarrow y=-\frac{6}{5}x+\frac{30}{5}\rightarrow y=-\frac{6}{5}x+6[/tex]So you got the slope right, it's 5/6.
Now, with the given point we find the intercept. The point is x = -6 and y = -7, so we replace these values into the expression we have until now:
[tex]y=\frac{5}{6}x+b[/tex][tex]-7=\frac{5}{6}(-6)+b[/tex]And solve for b
[tex]-7=-5+b\rightarrow b=-7+5=-2[/tex]So the equation of the line is:
[tex]y=\frac{5}{6}x-2[/tex]