a) \(\displaystyle{{5\left( {x - 1} \right) + 2} \over 6} - {{7x - 1} \over 4} \)\(\displaystyle = {{2\left( {2x + 1} \right)} \over 7} - 5\)
\(\displaystyle \Leftrightarrow {{5x - 5 + 2} \over 6} - {{7x - 1} \over 4} \) \(\displaystyle= {{4x + 2} \over 7} - 5 \)
\(\displaystyle \Leftrightarrow {{5x - 3} \over 6} - {{7x - 1} \over 4}\)\(\displaystyle = {{4x + 2} \over 7} - 5 \)
\( \Leftrightarrow \dfrac{14\left( {5x - 3} \right) - 21\left( {7x - 1} \right)}{84} \)\(\displaystyle= \dfrac {12\left( {4x + 2} \right) - 5.84}{84} \)
\( \Leftrightarrow 14\left( {5x - 3} \right) - 21\left( {7x - 1} \right) \)\(\displaystyle= 12\left( {4x + 2} \right) - 5.84 \)
\( \Leftrightarrow 70x - 42 - 147x + 21 \) \(\displaystyle= 48x + 24 - 420 \)
\( \Leftrightarrow 70x - 147x - 48x \) \(= 24 - 420 + 42 - 21 \)
\( \Leftrightarrow - 125x = - 375 \Leftrightarrow x = 3 \)
Vậy phương trình có nghiệm \(x = 3.\)
b) \(\displaystyle{{3\left( {x - 3} \right)} \over 4} + {{4x - 10,5} \over {10}} \) \(\displaystyle= {{3\left( {x + 1} \right)} \over 5} + 6\)
\(\displaystyle \Leftrightarrow {{3x - 9} \over 4} + {{4x - 10,5} \over {10}}\) \(\displaystyle = {{3x + 3} \over 5} + 6 \)
\(\Leftrightarrow \dfrac {5\left( {3x - 9} \right) + 2\left( {4x - 10,5} \right)}{20} \) \(= \dfrac {4\left( {3x + 3} \right) + 6.20}{20} \)
\(\Leftrightarrow 5\left( {3x - 9} \right) + 2\left( {4x - 10,5} \right) \) \(= 4\left( {3x + 3} \right) + 6.20 \)
\( \Leftrightarrow 15x - 45 + 8x - 21 \) \(= 12x + 12 + 120\)
\( \Leftrightarrow 15x + 8x - 12x \) \(= 12 + 120 + 45 + 21\)
\( \Leftrightarrow 11x = 198 \)
\( \Leftrightarrow x = 18 \)
Phương trình có nghiệm \(x = 18.\)
c) \(\displaystyle{{2\left( {3x + 1} \right) + 1} \over 4} - 5 \) \(\displaystyle= {{2\left( {3x - 1} \right)} \over 5} - {{3x + 2} \over {10}}\)
\(\displaystyle \Leftrightarrow {{6x + 2 + 1} \over 4} - 5\) \(\displaystyle = {{6x - 2} \over 5} - {{3x + 2} \over {10}} \)
\(\displaystyle \Leftrightarrow {{6x + 3} \over 4} - 5\) \(\displaystyle = {{6x - 2} \over 5} - {{3x + 2} \over {10}} \)
\(\displaystyle \Leftrightarrow \dfrac{5\left( {6x + 3} \right) - 5.20}{20} \) \(\displaystyle= \dfrac {4\left( {6x - 2} \right) - 2\left( {3x + 2} \right)}{20} \)
\(\displaystyle \Leftrightarrow 5\left( {6x + 3} \right) - 5.20 \) \(\displaystyle= 4\left( {6x - 2} \right) - 2\left( {3x + 2} \right) \)
\(\displaystyle \Leftrightarrow 30x + 15 - 100 \) \(\displaystyle= 24x - 8 - 6x - 4 \)
\(\Leftrightarrow 30x - 24x + 6x \) \(= - 8 - 4 - 15 + 100 \)
\(\displaystyle \Leftrightarrow 12x = 73 \Leftrightarrow x = {{73} \over {12}} \)
Phương trình có nghiệm \(\displaystyle x = {{73} \over {12}}\).
d) \(\displaystyle{{x + 1} \over 3} + {{3\left( {2x + 1} \right)} \over 4} \) \(\displaystyle= {{2x + 3\left( {x + 1} \right)} \over 6} + {{7 + 12x} \over {12}}\)
\(\displaystyle \Leftrightarrow {{x + 1} \over 3} + {{6x + 3} \over 4} \) \(\displaystyle = {{2x + 3x + 3} \over 6} + {{7 + 12x} \over {12}} \)
\(\displaystyle \Leftrightarrow {{x + 1} \over 3} + {{6x + 3} \over 4} \) \(\displaystyle= {{5x + 3} \over 6} + {{7 + 12x} \over {12}} \)
\(\displaystyle \Leftrightarrow \dfrac {4\left( {x + 1} \right) + 3\left( {6x + 3} \right)}{12} \) \(\displaystyle= \dfrac {2\left( {5x + 3} \right) + 7 + 12x}{12} \)
\(\displaystyle \Leftrightarrow 4\left( {x + 1} \right) + 3\left( {6x + 3} \right) \) \(\displaystyle= 2\left( {5x + 3} \right) + 7 + 12x \)
\( \Leftrightarrow 4x + 4 + 18x + 9\) \( = 10x + 6 + 7 + 12 x \)
\( \Leftrightarrow 4x + 18x - 10x-12x \) \(= 6 + 7 -4 - 9 \)
\( \Leftrightarrow 0x = 0\) (luôn đúng)
Vậy phương trình có vô số nghiệm.
