Question

How many pairs of integers $(b,c)$ satisfy the equation
\[\frac{b + 7}{b + 4} = \frac{c}{9}?\]

Answer 1

your notation clearly did not show up as you intended.

I will guess you meant:

(b+7)/(b+4) = c/9
c = 9(b+7)/(b+4)

I have this very old fashioned computer language called BASIC.
I ran a stupid little program for b = -5000 to b = 5000 and it found the following pairs (b,c)

>list
10 for b = -5000 to 5000
20 c = 9*(b+7)/(b+4)
30 if c = int(c) then print b,c
40 next b
>run
-31 8
-13 6
-7 0
-5 -18
-3 36
-1 18
5 12
23 10

So I found 8 of them

Answer 2

Don't knock BASIC. I knew a girl in college who was using it to program a gyroscopically stable table in orbit in 1972.