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The equation y = (x + A)/(Bx + C), where A,B, and C are integers, is shown below. What is A + B + C?

 

 May 19, 2022
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Note that when x = 4, y = 0. Then substituting (x, y) = (4, 0) gives:

\(\dfrac{4 + A}{4B + C} = 0\\ 4 + A = 0\\ A = -4\)

 

Now, note that the vertical asymptote occurs at x = 3. That means the denominator (Bx + C) is 0 when x = 3. Then 3B + C = 0.

 

Then, note that \(\displaystyle\lim_{x\to \infty} y = -1\) since y approaches -1 when x approaches infinity. Then \(\dfrac1B = -1\implies B = -1\).

 

Since 3B + C = 0, C = 3.

 

Therefore, the function is \(y = \dfrac{x - 4}{-x + 3}\). A + B + C = -4 + (-1) + 3 = -2.

 May 19, 2022

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