Phương trình $\dfrac{{x - 12}}{{77}} + \dfrac{{x - 11}}{{78}}$$= \dfrac{{x - 74}}{{15}} + \dfrac{{x - 73}}{{16}}$ có nghiệm là

Ta có $\dfrac{{x - 12}}{{77}} + \dfrac{{x - 11}}{{78}}$$= \dfrac{{x - 74}}{{15}} + \dfrac{{x - 73}}{{16}}$

$\Leftrightarrow \left( {\dfrac{{x - 12}}{{77}} - 1} \right) + \left( {\dfrac{{x - 11}}{{78}} - 1} \right)$$= \left( {\dfrac{{x - 74}}{{15}} - 1} \right) + \left( {\dfrac{{x - 73}}{{16}} - 1} \right)$

$\Leftrightarrow \left( {\dfrac{{x - 12 - 77}}{{77}}} \right) + \left( {\dfrac{{x - 11 - 78}}{{78}}} \right)$$= \left( {\dfrac{{x - 74 - 15}}{{15}}} \right) + \left( {\dfrac{{x - 73 - 16}}{{16}}} \right)$

$\Leftrightarrow \dfrac{{x - 89}}{{77}} + \dfrac{{x - 89}}{{78}} - \dfrac{{x - 89}}{{15}} - \dfrac{{x - 89}}{{16}} = 0$

$\Leftrightarrow \left( {x - 89} \right)\left( {\dfrac{1}{{77}} + \dfrac{1}{{78}} - \dfrac{1}{{15}} - \dfrac{1}{{16}}} \right) = 0$

Nhận thấy $\dfrac{1}{{77}} + \dfrac{1}{{78}} - \dfrac{1}{{15}} - \dfrac{1}{{16}} \ne 0$  nên $\left( {x - 89} \right)\left( {\dfrac{1}{{77}} + \dfrac{1}{{78}} - \dfrac{1}{{15}} - \dfrac{1}{{16}}} \right) = 0$$\Leftrightarrow x - 89 = 0 \Leftrightarrow x = 89$