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Giải các phương trình sau:
a. \({{5\left( {x – 1} \right) + 2} \over 6} – {{7x – 1} \over 4} = {{2\left( {2x + 1} \right)} \over 7} – 5\)
b. \({{3\left( {x – 3} \right)} \over 4} + {{4x – 10,5} \over {10}} = {{3\left( {x + 1} \right)} \over 5} + 6\)
c. \({{2\left( {3x + 1} \right) + 1} \over 4} – 5 = {{2\left( {3x – 1} \right)} \over 5} – {{3x + 2} \over {10}}\)
d. \({{x + 1} \over 3} + {{3\left( {2x + 1} \right)} \over 4} = {{2x + 3\left( {x + 1} \right)} \over 6} + {{7 + 12x} \over {12}}\)
a. \({{5\left( {x – 1} \right) + 2} \over 6} – {{7x – 1} \over 4} = {{2\left( {2x + 1} \right)} \over 7} – 5\)
\(\eqalign{ & \Leftrightarrow {{5x – 5 + 2} \over 6} – {{7x – 1} \over 4} = {{4x + 2} \over 7} – 5 \cr & \Leftrightarrow {{5x – 3} \over 6} – {{7x – 1} \over 4} = {{4x + 2} \over 7} – 5 \cr & \Leftrightarrow 14\left( {5x – 3} \right) – 21\left( {7x – 1} \right) = 12\left( {4x + 2} \right) – 5.84 \cr & \Leftrightarrow 70x – 42 – 147x + 21 = 48x + 24 – 420 \cr & \Leftrightarrow 70x – 147x – 48x = 24 – 420 + 42 – 21 \cr & \Leftrightarrow – 125x = – 375 \cr & \Leftrightarrow x = 3 \cr} \)
Phương trình có nghiệm x = 3
b. \({{3\left( {x – 3} \right)} \over 4} + {{4x – 10,5} \over {10}} = {{3\left( {x + 1} \right)} \over 5} + 6\)
\(\eqalign{ & \Leftrightarrow {{3x – 9} \over 4} + {{4x – 10,5} \over {10}} = {{3x + 3} \over 5} + 6 \cr & \Leftrightarrow 5\left( {3x – 9} \right) + 2\left( {4x – 10,5} \right) = 4\left( {3x + 3} \right) + 6.20 \cr & \Leftrightarrow 15x – 45 + 8x – 21 = 12x + 12 + 120 & \Leftrightarrow 15x + 8x – 12x = 12 + 120 + 45 + 21 \cr & \Leftrightarrow 11x = 198 \cr & \Leftrightarrow x = 18 \cr} \)
Phương trình có nghiệm x = 18
c. \({{2\left( {3x + 1} \right) + 1} \over 4} – 5 = {{2\left( {3x – 1} \right)} \over 5} – {{3x + 2} \over {10}}\)
\(\eqalign{ & \Leftrightarrow {{6x + 2 + 1} \over 4} – 5 = {{6x – 2} \over 5} – {{3x + 2} \over {10}} \cr & \Leftrightarrow {{6x + 3} \over 4} – 5 = {{6x – 2} \over 5} – {{3x + 2} \over {10}} \cr & \Leftrightarrow 5\left( {6x + 3} \right) – 5.20 = 4\left( {6x – 2} \right) – 2\left( {3x + 2} \right) \cr & \Leftrightarrow 30x + 15 – 100 = 24x – 8 – 6x – 4 \cr & \Leftrightarrow 30x – 24x + 6x = – 8 – 4 – 15 + 100 \cr & \Leftrightarrow 12x = 73 \Leftrightarrow x = {{73} \over {12}} \cr} \)
Phương trình có nghiệm \(x = {{73} \over {12}}\)
d. \({{x + 1} \over 3} + {{3\left( {2x + 1} \right)} \over 4} = {{2x + 3\left( {x + 1} \right)} \over 6} + {{7 + 12x} \over {12}}\)
\(\eqalign{ & \Leftrightarrow {{x + 1} \over 3} + {{6x + 3} \over 4} = {{2x + 3x + 3} \over 6} + {{7 + 12x} \over {12}} \cr & \Leftrightarrow {{x + 1} \over 3} + {{6x + 3} \over 4} = {{5x + 3} \over 6} + {{7 + 12x} \over {12}} \cr & \Leftrightarrow 4\left( {x + 1} \right) + 3\left( {6x + 3} \right) = 2\left( {5x + 3} \right) + 7 + 12x \cr & \Leftrightarrow 4x + 4 + 18x + 9 = 10x + 6 + 7 + 12 \cr & \Leftrightarrow 4x + 18x – 10x = 6 + 7 + 12 – 9 \cr & \Leftrightarrow 0x = 0 \cr} \)
Phương trình có vô số nghiệm.