Giải các phương trình: a) x+1 / 2024 + x+2 / 2023 = x+3 / 2022 + x+4 / 2021

Hướng dẫn giải

a) \[\frac{{x + 1}}{{2024}} + \frac{{x + 2}}{{2023}} = \frac{{x + 3}}{{2022}} + \frac{{x + 4}}{{2021}}\]

\[\left( {\frac{{x + 1}}{{2024}} + 1} \right) + \left( {\frac{{x + 2}}{{2023}} + 1} \right) = \left( {\frac{{x + 3}}{{2022}} + 1} \right) + \left( {\frac{{x + 4}}{{2021}} + 1} \right)\]

\[\frac{{x + 2025}}{{2024}} + \frac{{x + 2025}}{{2023}} = \frac{{x + 2025}}{{2022}} + \frac{{x + 2025}}{{2021}}\]

\[\frac{{x + 2025}}{{2024}} + \frac{{x + 2025}}{{2023}} - \frac{{x + 2025}}{{2022}} - \frac{{x + 2025}}{{2021}} = 0\]

\[\left( {x + 2025} \right)\left( {\frac{1}{{2024}} + \frac{1}{{2023}} - \frac{1}{{2022}} - \frac{1}{{2021}}} \right) = 0\]

Vì \[\frac{1}{{2024}} < \frac{1}{{2022}}\] nên \[\frac{1}{{2024}} - \frac{1}{{2022}} < 0\].

Vì \[\frac{1}{{2023}} < \frac{1}{{2021}}\] nên \[\frac{1}{{2023}} - \frac{1}{{2021}} < 0\].

Do đó \[\frac{1}{{2024}} + \frac{1}{{2023}} - \frac{1}{{2022}} - \frac{1}{{2021}} < 0\] hay \[\frac{1}{{2024}} + \frac{1}{{2023}} - \frac{1}{{2022}} - \frac{1}{{2021}} \ne 0\].

Khi đó \[x + 2025 = 0\] nên \[x = - 2025\].

Vậy nghiệm của phương trình là \[x = - 2025\].

b) \[\frac{{2027 - x}}{{73}} + \frac{{2025 - x}}{{75}} + \frac{{2023 - x}}{{77}} + \frac{{2021 - x}}{{79}} + 4 = 0\]

\[\left( {\frac{{2027 - x}}{{73}} + 1} \right) + \left( {\frac{{2025 - x}}{{75}} + 1} \right) + \left( {\frac{{2023 - x}}{{77}} + 1} \right) + \left( {\frac{{2021 - x}}{{79}} + 1} \right) = 0\]

\[\frac{{2100 - x}}{{73}} + \frac{{2100 - x}}{{75}} + \frac{{2100 - x}}{{77}} + \frac{{2100 - x}}{{79}} = 0\]

\[\left( {2100 - x} \right)\left( {\frac{1}{{73}} + \frac{1}{{75}} + \frac{1}{{77}} + \frac{1}{{79}}} \right) = 0\]

\[2100 - x = 0\] (vì \[\frac{1}{{73}} + \frac{1}{{75}} + \frac{1}{{77}} + \frac{1}{{79}} > 0\])

\[x = 2100\]

Vậy nghiệm của phương trình là \[x = 2100\].