Найдите ab/c, если ab / (a+b) = 1; ac / (a+c) = 2; bc / (b+c) = 3.

Найдем a * b/c, если a * b / (a + b) = 1; a * c / (a + c) = 2; b * c / (b + c) = 3.

Решим систему уравнений:

{ a * b / (a + b) = 1;

a * c / (a + c) = 2;

b * c / (b + c) = 3;

1) a * b / (a + b) = 1;

a * b = a + b;

a * b - a = b;

a * (b - 1) = b;

a = b / (b - 1);

2) a * c / (a + c) = 2;

a * c = 2 * a + 2 * c;

a * c - 2 * c = 2 * a;

c = 2 * a / (a - 2);

c = 2 * (b / (b - 1)) / (b / (b - 1) - 2);

c = 2 * b / (b - 1) / ((b - 2 * (b - 1)) / (b - 1));

c = 2 * b / (b - 1) / ((b - 2 * b + 2) / (b - 1));

c = 2 * b / (b - 1) * (b - 1) / (2 - b);

c = 2 * b / (2 - b);

3) b * c / (b + c) = 3;

(b * 2 * b / (2 - b)) / (b + 2 * b / (2 - b)) = 3;

(2 * b^2 / (2 - b)) / ((b * (2 - b) + 2 * b) / (2 - b)) = 3;

(2 * b^2 / (2 - b)) / ((2 * b - b^2 + 2 * b) / (2 - b)) = 3;

(2 * b^2 / (2 - b)) * (2 - b) / (4 * b - b^2) = 3;

2 * b^2 / (4 * b * b^2) = 3;

2 * b / (4 - b) = 3;

3 * (4 - b) = 2 * b;

12 - 3 * b = 2 * b;

12 = 5 * b;

b = 2.4;

4) a = b / (b - 1) = 2.4 / (2.4 - 1) = 2.4/1.4 = 24/14 = 12/7;

5) c = 2 * b / (2 - b) = 2 * 12/7 / (2 - 12/7) = 24/7 / (2/7) = 24/7 * 7/2 = 24/2 = 12.

Теперь найдем значение выражения a * b/c:

a * b/c = 12/7 * 2.4/12 = 12 * 2.4 / (7 * 12) = 2.4/7 = (2 4/10) / 7 = (2 2/5) / 7 = (12/5) / 7 = 12/5 * 1/7 = 12/35.