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Let \(y = \frac{x - 3}{x + 7}\). Then this equation can be expressed in the form \((x + a)(y + b) = c\) for some constants \(a, b,\) and \(c.\) Enter your answer in the form "\(a, b, c\)".

 Apr 19, 2023
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To write the given equation in the form (x + a)(y + b) = c, we need to expand the right-hand side and match the terms with the given equation.

(x + a)(y + b) = c

Expanding the left-hand side, we get:

xy + xb + ay + ab = c

Comparing the coefficients with the given equation, we have:

ab = 3 (coefficient of x in numerator) a + b = -4 (coefficient of x in denominator) c = -21 (constant term)

Solving for a and b, we can use the fact that a + b = -4 to express one variable in terms of the other:

a = -4 - b

Substituting into the equation ab = 3, we have:

(-4 - b) * b = 3

Expanding and solving for b, we get:

b^2 + 4b + 3 = 0

(b + 3)(b + 1) = 0

So, b = -3 or b = -1.

If b = -3, then a = -4 - (-3) = -1. This gives us:

(x - 1)(y - 3) = -21

If b = -1, then a = -4 - (-1) = -3. This gives us:

(x - 3)(y - 1) = -21

Therefore, the equation y = (x - 3)/(x + 7) can be expressed in the form:

(-1, -3, -21) or (-3, -1, -21)

 Apr 19, 2023

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