Giải phương trình: (x - 10_ / 1994 + (x + 8) / 1996 + (x - 6) / 1998+ (x - 4) / 2000
Câu hỏi:
Giải phương trình:
\(\frac{{x - 10}}{{1994}} + \frac{{x - 8}}{{1996}} + \frac{{x - 6}}{{1998}} + \frac{{x - 4}}{{2000}} + \frac{{x - 2}}{{2002}} = \frac{{x - 2002}}{2} + \frac{{x - 2000}}{4} + \frac{{x - 1998}}{6} + \frac{{x - 1996}}{8} + \frac{{x - 1994}}{{10}}\)
Trả lời:
\(\frac{{x - 10}}{{1994}} + \frac{{x - 8}}{{1996}} + \frac{{x - 6}}{{1998}} + \frac{{x - 4}}{{2000}} + \frac{{x - 2}}{{2002}} = \frac{{x - 2002}}{2} + \frac{{x - 2000}}{4} + \frac{{x - 1998}}{6} + \frac{{x - 1996}}{8} + \frac{{x - 1994}}{{10}}\)
⇔ \(\left( {\frac{{x - 10}}{{1994}} - 1} \right) + \left( {\frac{{x - 8}}{{1996}} - 1} \right) + \left( {\frac{{x - 6}}{{1998}} - 1} \right) + \left( {\frac{{x - 4}}{{2000}} - 1} \right) + \left( {\frac{{x - 2}}{{2002}} - 1} \right)\)
\( = \left( {\frac{{x - 2002}}{2} - } \right)1 + \left( {\frac{{x - 2000}}{4} - 1} \right) + \left( {\frac{{x - 1998}}{6} - 1} \right) + \left( {\frac{{x - 1996}}{8} - 1} \right) + \left( {\frac{{x - 1994}}{{10}} - 1} \right)\)
⇔ \(\frac{{x - 2014}}{{1994}} + \frac{{x - 2014}}{{1996}} + \frac{{x - 2014}}{{1998}} + \frac{{x - 2014}}{{2000}} + \frac{{x - 2014}}{{2002}}\)
\( = \frac{{x - 2014}}{2} + \frac{{x - 2014}}{4} + \frac{{x - 2014}}{6} + \frac{{x - 2014}}{8} + \frac{{x - 2014}}{{10}}\)
⇔ (x – 2014)\[\left( {\frac{1}{{1994}} + \frac{1}{{1996}} + \frac{1}{{1998}} + \frac{1}{{2000}} + \frac{1}{{2002}} - \frac{1}{2} - \frac{1}{4} - \frac{1}{6} - \frac{1}{8} - \frac{1}{{10}}} \right) = 0\]
⇔ x – 2014 = 0
⇔ x = 2014.
Vậy x = 2014.
