Phương trình $\dfrac{{x - 2}}{{77}} + \dfrac{{x - 1}}{{78}} = \dfrac{{x - 74}}{5} + \dfrac{{x - 73}}{6}$ có nghiệm là:

Ta có:

$\dfrac{{x - 2}}{{77}} + \dfrac{{x - 1}}{{78}} = \dfrac{{x - 74}}{5} + \dfrac{{x - 73}}{6}$$\Leftrightarrow \left( {\dfrac{{x - 2}}{{77}} - 1} \right) + \left( {\dfrac{{x - 1}}{{78}} - 1} \right)$$= \left( {\dfrac{{x - 74}}{5} - 1} \right) + \left( {\dfrac{{x - 73}}{6} - 1} \right)$

$\Leftrightarrow \dfrac{{x - 79}}{{77}} + \dfrac{{x - 79}}{{78}} = \dfrac{{x - 79}}{5} + \dfrac{{x - 79}}{6}$$\Leftrightarrow \dfrac{{x - 79}}{{77}} + \dfrac{{x - 79}}{{78}} - \dfrac{{x - 79}}{5} - \dfrac{{x - 79}}{6} = 0$

$\Leftrightarrow \left( {x - 79} \right)\left( {\dfrac{1}{{77}} + \dfrac{1}{{78}} - \dfrac{1}{5} - \dfrac{1}{6}} \right) = 0$$\Leftrightarrow x - 79 = 0 \Leftrightarrow x = 79$

(vì $\dfrac{1}{{77}} < \dfrac{1}{5},\dfrac{1}{{78}} < \dfrac{1}{6}$ nên $\dfrac{1}{{77}} + \dfrac{1}{{78}} - \dfrac{1}{5} - \dfrac{1}{6} < 0$)

Vậy phương trình có nghiệm duy nhất $x = 79$.