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# help

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Solve for x: a/(x - a) + b/(x - b) = 2c/(x - c).

Oct 26, 2019

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Solve for x:
a/(x - a) + b/(x - b) = (2 c)/(x - c)

Multiply both sides by (x - a) (x - c):
-a (c - x) - (b (x - a) (x - c))/(b - x) = -2 c (a - x)

Bring -a (c - x) - (b (x - a) (x - c))/(b - x) together using the common denominator x - b:
(2 a b c - 2 a b x - a c x - b c x + a x^2 + b x^2)/(x - b) = -2 c (a - x)

Multiply both sides by x - b:
2 a b c - 2 a b x - a c x - b c x + a x^2 + b x^2 = -2 c (a - x) (x - b)

Collect in terms of x:
2 a b c + x (-2 a b - a c - b c) + x^2 (a + b) = -2 c (a - x) (x - b)

Expand and collect in terms of x:
2 a b c + x (-2 a b - a c - b c) + x^2 (a + b) = 2 a b c + x (-2 a c - 2 b c) + 2 c x^2

Subtract 2 a b c + (-2 a c - 2 b c) x + 2 c x^2 from both sides:
-x (-2 a c - 2 b c) + x (-2 a b - a c - b c) + x^2 (a + b) - 2 c x^2 = 0

Expand and collect in terms of x:
x (-2 a b + a c + b c) + x^2 (a + b - 2 c) = 0

Factor x and constant terms from the left hand side:
-x (2 a b - a c - b c - a x - b x + 2 c x) = 0

Multiply both sides by -1:
x (2 a b - a c - b c - a x - b x + 2 c x) = 0

Split into two equations:
x = 0 or 2 a b - a c - b c - a x - b x + 2 c x = 0

Collect in terms of x:
x = 0 or 2 a b - a c - b c + x (-a - b + 2 c) = 0

Subtract 2 a b - a c - b c from both sides:
x = 0 or x (2 c + (-a - b)) = -2 a b + a c + b c

Divide both sides by -a - b + 2 c:

x = 0           or               x = (-2 a b + a c + b c)/(-a - b + 2 c)

Oct 26, 2019