Question

a/x-a + b/x-b =2c/x-c find x

Answer 1

The value of x is $\frac{2ab - ac - bc}{a+b - 2c}$.

Step-by-step explanation:

$\frac{a}{(x-a)} +\frac{b}{(x-b)} = \frac{2c}{(x-c)} \\=> \frac{a(x-b) + b(x-a)}{(x-a)(x-b)} = \frac{2c}{(x-c)} \\=> \frac{ax - ab + bx - ab}{x^{2}-bx-ax+ab } = \frac{2c}{(x-c)} \\=> \frac{ax +bx-2ab}{x^{2}-ax-bx+ab } = \frac{2c}{(x-c)}$

=> (ax + bx - 2ab)(x - c) = 2c(x² - ax - bx +ab)

=> ax² + bx² - 2abx - acx - bcx + 2abc = 2cx² - 2acx - 2bcx + 2abc

=> ax² + bx² - 2cx² - 2abx + acx + bcx = 0

=> ax² + bx² - 2cx² = 2abx - acx - bcx

=> x²(a + b - 2c) = x(2ab - ac - bc)

=> x(a + b - 2c) = 2ab - ac - bc

=> x = $\frac{2ab - ac - bc}{a+b - 2c}$

Hence, the value of x is $\frac{2ab - ac - bc}{a+b - 2c}$.

Answer 2

$x=\frac{2ab-bc-ac}{(a+b-2c)}$

Step-by-step explanation:

$\frac{a}{x-a} +\frac{b}{x-b} =\frac{2c}{x-c}$

$\Leftrightarrow \frac{a}{x-a} +\frac{b}{x-b} =\frac{c}{x-c} +\frac{c}{x-c}$

$\Leftrightarrow \frac{a}{x-a} -\frac{c}{x-c} =\frac{c}{x-c}-\frac{b}{x-b}$

$\Leftrightarrow \frac{a(x-c)-c(x-a)}{(x-a)(x-c)} =\frac{c(x-b)-b(x-c)}{(x-c)(x-b)}$

$\Leftrightarrow \frac{ax-ac-cx+ca}{(x-a)(x-c)} =\frac{cx-cb-bx+bc}{(x-c)(x-b)}$

$\Leftrightarrow \frac{ax-cx}{(x-a)(x-c)} =\frac{cx-bx}{(x-c)(x-b)}$

$\Leftrightarrow \frac{x(a-c)}{(x-a)} =\frac{x(c-b)}{(x-b)}$

$\Leftrightarrow \frac{(a-c)}{(x-a)} =\frac{(c-b)}{(x-b)}$

$\Leftrightarrow{(a-c)}{{(x-b)} ={(c-b)}(x-a)$

$\Leftrightarrow ax-ab-cx+bc=cx-ac-bx+ab$

$\Leftrightarrow ax-ab-cx+bc-cx+ac+bx-ab=0$

$\Leftrightarrow x(a+b-2c)=2ab-bc-ac$

$\Leftrightarrow x=\frac{2ab-bc-ac}{(a+b-2c)}$