# Question #cf38f

Mar 17, 2017

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{x - 3}{x - 4} - \frac{x - 7}{x - 8}$

$\implies \frac{x - 2 + 1}{x - 2} - \frac{x - 6 + 1}{x - 6} = \frac{x - 4 + 1}{x - 4} - \frac{x - 8 + 1}{x - 8}$

$\implies 1 + \frac{1}{x - 2} - 1 - \frac{1}{x - 6} = 1 + \frac{1}{x - 4} - 1 - \frac{1}{x - 8}$

$\implies \frac{1}{x - 2} - \frac{1}{x - 6} = \frac{1}{x - 4} - \frac{1}{x - 8}$

$\implies \frac{x - 6 - x + 2}{\left(x - 2\right) \left(x - 6\right)} = \frac{x - 8 - x + 4}{\left(x - 4\right) \left(x - 8\right)}$

$\implies - \frac{4}{\left(x - 2\right) \left(x - 6\right)} = - \frac{4}{\left(x - 4\right) \left(x - 8\right)}$

$\implies \left(x - 2\right) \left(x - 6\right) = \left(x - 4\right) \left(x - 8\right)$

$\implies {x}^{2} - 8 x + 12 = {x}^{2} - 12 x + 32$

$\implies 12 x - 8 x = 32 - 12 = 20$

$\implies 4 x = 20$

$\implies x = \frac{20}{4} = 5$

Mar 17, 2017

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{\left(x - 1\right) \left(x - 6\right) - \left(x - 2\right) \left(x - 5\right)}{\left(x - 2\right) \left(x - 6\right)} = \frac{{x}^{2} - 6 x - x + 6 - {x}^{2} + 5 x + 2 x - 10}{\left(x - 2\right) \left(x - 6\right)} = - \frac{4}{\left(x - 2\right) \left(x - 6\right)}$

#### Explanation:

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{\left(x - 1\right) \left(x - 6\right) - \left(x - 2\right) \left(x - 5\right)}{\left(x - 2\right) \left(x - 6\right)} = \frac{{x}^{2} - 6 x - x + 6 - {x}^{2} + 5 x + 2 x - 10}{\left(x - 2\right) \left(x - 6\right)} = - \frac{4}{\left(x - 2\right) \left(x - 6\right)}$
$\frac{x - 3}{x - 4} - \frac{x - 7}{x - 8} = \frac{\left(x - 3\right) \left(x - 8\right) - \left(x - 4\right) \left(x - 7\right)}{\left(x - 4\right) \left(x - 8\right)} = \frac{{x}^{2} - 8 x - 3 x + 24 - {x}^{2} + 7 x + 4 x - 28}{\left(x - 4\right) \left(x - 8\right)} = - \frac{4}{\left(x - 4\right) \left(x - 8\right)}$
So you get
$\left(x - 2\right) \left(x - 6\right) = \left(x - 4\right) \left(x - 8\right)$
$4 x = 20.$
$x = 5.$