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

* Hướng dẫn giải

Ta có:

 \(\begin{array}{l} \frac{{x - 2}}{{77}} + \frac{{x - 1}}{{78}} = \frac{{x - 74}}{5} + \frac{{x - 73}}{6}\\ \frac{{x - 2}}{{77}} - 1 + \frac{{x - 1}}{{78}} - 1 = \frac{{x - 74}}{5} - 1 + \frac{{x - 73}}{6} - 1\\ \Leftrightarrow \frac{{x - 79}}{{77}} + \frac{{x - 79}}{{78}} = \frac{{x - 79}}{5} + \frac{{x - 79}}{6}\\ \Leftrightarrow \frac{{x - 79}}{{77}} + \frac{{x - 79}}{{78}} - \frac{{x - 79}}{5} - \frac{{x - 79}}{6} = 0\\ \Leftrightarrow \left( {x - 79} \right)\left( {\frac{1}{{77}} + \frac{1}{{78}} - \frac{1}{5} - \frac{1}{6}} \right) = 0\\ \Leftrightarrow x - 79 = 0 \Leftrightarrow x = 79 \end{array}\)

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

Vậy phương trình có nghiệm duy nhất: x=79