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# PLS HALP

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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