What is the slope of a line perpendicular to 8x+9y=1

The slope of a line that is perpendicular is the negative reciprocal.

In slope intercept form the line you gave is y = - 8/9x+1/9.

So the slope is - 8/9.

Meaning the slope of a line that is perpendicular would be 9/8.

If put into slope intercept form ... y=mx+b where m=slope

8x+9y=1

9y=-8x+1

y = (-8x+1) / 9

So the slope is - 8/9

For a line to be perpendicular to this one, it must have the negative reciprocal slope of the reference line:

Mathematically, for two lines to be perpendicular to each other:

m1*m2=-1 in this case:

-8m/9=-1

m=-9/-8

m=9/8 or if you prefer

m=1.125

For your curiosity ... since slope equals dy/dx we can say that:

tanα = (dy/dx) and in this context

tanα=m so the angle is:

α=arctan (m)

a=arctan (-8/9) ≈-41.63

b=arctan (9/8) ≈48.37

So the difference between b and a is 48.37--41.63=90° Since they are at a 90° to each other, they are perpendicular ...