how many ordered pair of a,b integers are there with \[\frac{a + 2}{a + 5} = \frac{b}{3}.\]
+2666 Cross multiply: \(3(a+2) = b(a+5)\)
Simplify: \(3a + 6 = ab + 5b\)
Isolate the constant: \(3a -ab-5b= -6\)
Factor out a: \(a(3-b) - 5b = -6\)
Add 15 to both sides: \(a(3-b) - 5b + 15 = 9\)
Factor out b: \(a(3-b) + 5 (3 - b) = 9\)
Combine like terms: \((a+5)(3-b) = 9\)
Note that \(a + 5\) must equal 1, 3, or 9, so \(3- b\) can equal the other factor of 9.
Thus, there are \(\color{brown}\boxed{3}\) cases that work.
Note that the cases are (4, 2), (-2, 0), and (-4, -6)
