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how many ordered pair of a,b integers are there with \[\frac{a + 2}{a + 5} = \frac{b}{3}.\]

 Jun 26, 2022
 #1
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Do not post A o P S homework.

 Jun 26, 2022
 #2
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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)

 Jun 26, 2022

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