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Giải các phương trình sau:
a. \({{2x} \over 3} + {{2x – 1} \over 6} = 4 – {x \over 3}\)
b.\({{x – 1} \over 2} + {{x – 1} \over 4} = 1 – {{2\left( {x – 1} \right)} \over 3}\)
c. \({{2 – x} \over {2001}} – 1 = {{1 – x} \over {2002}} – {x \over {2003}}\)
a. \({{2x} \over 3} + {{2x – 1} \over 6} = 4 – {x \over 3}\)
\( \Leftrightarrow 2.2x + 2x – 1 = 4.6 – 2x\)
\(\eqalign{ & \Leftrightarrow 4x + 2x – 1 = 24 – 2x \Leftrightarrow 6x + 2x = 24 + 1 \cr & \Leftrightarrow 8x = 25 \Leftrightarrow x = {{25} \over 8} \cr} \)
Phương trình có nghiệm $x = {{25} \over 8}\)
b. \({{x – 1} \over 2} + {{x – 1} \over 4} = 1 – {{2\left( {x – 1} \right)} \over 3}\)
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\( \Leftrightarrow {{x – 1} \over 2} + {{x – 1} \over 4} = 1 – {{2x – 2} \over 3}\)
\(\eqalign{ & \Leftrightarrow 6\left( {x – 1} \right) + 3\left( {x – 1} \right) = 12 – 4\left( {2x – 2} \right) \cr & \Leftrightarrow 6x – 6 + 3x – 3 = 12 – 8x + 8 \cr & \Leftrightarrow 6x + 3x + 8x = 12 + 8 + 6 + 3 \cr & \Leftrightarrow 17x = 29 \Leftrightarrow x = {{29} \over {17}} \cr} \)
Phương trình có nghiệm $x = {{29} \over {17}}\)
c. \({{2 – x} \over {2001}} – 1 = {{1 – x} \over {2002}} – {x \over {2003}}\)
\( \Leftrightarrow {{2 – x} \over {2001}} – 1 + 2 = {{1 – x} \over {2002}} + 1 + 1 – {x \over {2003}}\)
\(\eqalign{ & \Leftrightarrow {{2 – x} \over {2001}} + 1 = \left( {{{1 – x} \over {2002}} + 1} \right) + \left( {1 – {x \over {2003}}} \right) \cr & \Leftrightarrow {{2003 – x} \over {2001}} = {{2003 – x} \over {2002}} + {{2003 – x} \over {2003}} \cr & \Leftrightarrow {{2003 – x} \over {2001}} – {{2003 – x} \over {2002}} – {{2003 – x} \over {2003}} = 0 \cr & \Leftrightarrow \left( {2003 – x} \right)\left( {{1 \over {2001}} – {1 \over {2002}} – {1 \over {2003}}} \right) = 0 \cr & \Leftrightarrow 2003 – x = 0 \Leftrightarrow x = 2003 \cr} \)
Phương trình có nghiệm x = 2003.