1. Ví dụ

Ví dụ 1: Giải các phương trình sau:

a/ $\dfrac{x+4}{4}-\dfrac{x-3}{6}=\dfrac{x}{3}$
b/ $\dfrac{x-1}{2}-\dfrac{1-x}{4}=1-\dfrac{2(x-1)}{3}$
c/ $\dfrac{3 x-2}{6}-5=\dfrac{3-2(x+7)}{4}$
d/ $\dfrac{4 x+1}{3}-\dfrac{2}{3}-\dfrac{x-3}{6}=x$

Giải

a/ $\dfrac{x+4}{4}-\dfrac{x-3}{6}=\dfrac{x}{3} $

$ \Leftrightarrow \dfrac{3(x+4)-2(x-3)}{12} =\dfrac{4 x}{12} $

$ \Leftrightarrow 3x+12-2x+6=4x $
$ \Leftrightarrow -3x = -18 $
$ \Leftrightarrow x = 6 $
Vậy $ S= \{ 6 \} $

 

b/ $\dfrac{x-1}{2}-\dfrac{1-x}{4}=1-\dfrac{2(x-1)}{3}$

$ \Leftrightarrow (x-1) \left (\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{2}{3} \right)=1 $

$ \Leftrightarrow \dfrac{17}{12}(x-1)=1 $

$ \Leftrightarrow x-1 = \dfrac{12}{17} $

$ \Leftrightarrow x= \dfrac{12}{17}+1 = \dfrac{29}{17} $

Vậy $ S = \left \{ \dfrac{29}{17} \right \} $

 

c/ $\dfrac{3 x-2}{6}-5=\dfrac{3-2(x+7)}{4}$

$ \Leftrightarrow \dfrac{2(3x-2)-60}{12}= \dfrac{3[3-2x(x+7)]}{12} $

$ \Leftrightarrow 6x-4 -60 = 9-6x-42 $
$ \Leftrightarrow 12x = 31 $
$\Leftrightarrow x= \dfrac{31}{12} $

Vậy $ S= \left \{ \dfrac{31}{12} \right \} $

 

d/ $\dfrac{4 x+1}{3}-\dfrac{2}{3}-\dfrac{x-3}{6}=x$

$ \Leftrightarrow \dfrac{2(4x+1)-2 \cdot 2- (x-3)}{6}= \dfrac{6x}{6} $

$ \Leftrightarrow 8x+2-4 -x +3 = 6x $
$ \Leftrightarrow x = -1 $
Vậy $ S = \{ -1 \} $

Ví dụ 2: Giải các phương trình sau:

a/ $\dfrac{x}{2000}+\dfrac{x+1}{2001}+\dfrac{x+2}{2002}+\dfrac{x+3}{2003}=4$

b/ $\dfrac{59-x}{41}+\dfrac{57-x}{43}+\dfrac{55-x}{45}+d\dfrac{53-x}{47}+\dfrac{51-x}{49}=-5$

c/ $\dfrac{x+14}{86}+\dfrac{x+15}{85}+\dfrac{x+16}{84}+\dfrac{x+17}{83}+\dfrac{x+116}{4}=0$

d/ $\dfrac{x-90}{10}+\dfrac{x-76}{12}+\dfrac{x-58}{14}+\dfrac{x-36}{16}+\dfrac{x-15}{17}=15$

Giải

a/ $\dfrac{x}{2000}+\dfrac{x+1}{2001}+\dfrac{x+2}{2002}+\dfrac{x+3}{2003}=4$

$ \Leftrightarrow \left (\dfrac{x}{2000}-1 \right) + \left (\dfrac{x+1}{2001}-1 \right) + \left (\dfrac{x+2}{2002}-1 \right)+\left (\dfrac{x+3}{2003}-1 \right) =0 $

$ \Leftrightarrow \dfrac{x-2000}{2000}+\dfrac{x-2000}{2001} + \dfrac{x-2000}{2002}+\dfrac{x-2000}{2003} = 0 $

$ \Leftrightarrow (x-2000) \left(\dfrac{1}{2000}+\dfrac{1}{2001}+\dfrac{1}{2002}+\dfrac{1}{2003} \right) = 0 $

$ \Leftrightarrow x-2000 = 0 $
$\Leftrightarrow x = 2000$
Vậy $ S = \{ 2000 \} $

 

b/ $\dfrac{59-x}{41}+\dfrac{57-x}{43}+\dfrac{55-x}{45}+\dfrac{53-x}{47}+\dfrac{51-x}{49}=-5$

$ \Leftrightarrow \left(\dfrac{59-x}{41}+1 \right) +\left(\dfrac{57-x}{43}+1 \right)+\left(\dfrac{55-x}{45}+1 \right) +\left (\dfrac{53-x}{47}+1 \right) +\left(\dfrac{51-x}{49}+1 \right) = 0$

$ \Leftrightarrow \dfrac{100-x}{41}+\dfrac{100-x}{43}+\dfrac{100-x}{45}+\dfrac{100-x}{47}+\dfrac{1900-x}{49} = 0 $

$\Leftrightarrow (100-x) \left(\dfrac{1}{41}+\dfrac{1}{43}+\dfrac{1}{45}+\dfrac{1}{47}+\dfrac{1}{49} \right) = 0 $

$\Leftrightarrow 100 – x = 0 $
$\Leftrightarrow x = 100 $
Vậy $ S = \{ 100 \} $

c/ $\dfrac{x+14}{86}+\dfrac{x+15}{85}+\dfrac{x+16}{84}+\dfrac{x+17}{83}+\dfrac{x+116}{4}=0$

$\Leftrightarrow \left(\dfrac{x+14}{86}+1 \right)+\left(\dfrac{x+15}{85}+1 \right)+\left(\dfrac{x+16}{84}+1 \right)+\left(\dfrac{x+17}{83}+1 \right)+\left(\dfrac{x+116}{4}-4 \right)=0 $

$\Leftrightarrow \dfrac{x+100}{86}+\dfrac{x+100}{85}+\dfrac{x+100}{84}+\dfrac{x+100}{83}+\dfrac{x+100}{4} = 0 $

$\Leftrightarrow (x+100) \left(\dfrac{1}{86}+\dfrac{1}{85}+\dfrac{1}{84}+\dfrac{1}{83}+\dfrac{1}{4} \right) = 0 $

$\Leftrightarrow (x+100) = 0 $
$\Leftrightarrow x = – 100 $
Vậy $ S = \{ -100 \} $

 

d/ $\dfrac{x-90}{10}+\dfrac{x-76}{12}+\dfrac{x-58}{14}+\dfrac{x-36}{16}+\dfrac{x-15}{17}=15$

$\Leftrightarrow \left(\dfrac{x-90}{10}-1 \right)+\left(\dfrac{x-76}{12}-2 \right)+\left(\dfrac{x-58}{14}-3 \right)+\left(\dfrac{x-36}{16}-4 \right)+\left(\dfrac{x-15}{17}-5 \right) =0 $

$\Leftrightarrow \dfrac{x-100}{10}+\dfrac{x-100}{12}+\dfrac{x-100}{14}+\dfrac{x-100}{16}+\dfrac{x-100}{17} = 0 $

$\Leftrightarrow (x-100) \left(\dfrac{1}{86}+\dfrac{1}{10}+\dfrac{1}{12}+\dfrac{1}{14}+\dfrac{1}{16}+ \dfrac{1}{17} \right) = 0 $

$\Leftrightarrow (x-100) = 0 $
$\Leftrightarrow x = 100 $
Vậy $ S = \{ 100 \} $

2. Bài tập tự luyện

Bài 1: Giải các phương trình sau:

a/ $ \dfrac{x-2}{4}=\dfrac{5x+7}{8} $

b/ $ \dfrac{3x-2}{5}=\dfrac{4-7x}{3} $

c/ $ 1+ \dfrac{x}{9}= \dfrac{4}{3} $

d/ $ \dfrac{2x}{3}-\dfrac{2x-5}{6} = \dfrac{1}{2} $

e/ $ \dfrac{5x+2}{6}-x=1- \dfrac{x+2}{3} $

f/ $ 2x-\dfrac{1}{2}=\dfrac{2x+1}{4}-\dfrac{1-2x}{8} $

Bài 2: Giải các phương trình sau:

a/ $ \dfrac{x+3}{4}+2x-1 = \dfrac{x}{2} -\dfrac{x+2}{3} $

b/ $ \dfrac{5x-1}{10}+\dfrac{2x+3}{6}=\dfrac{x-8}{15}-\dfrac{x}{30} $

c/ $ \dfrac{(3x-1)(x+1)}{2}-\dfrac{3x^2}{2} = \dfrac{x-2}{2} $

d/ $ \dfrac{2(x+5)}{3}+\dfrac{x+12}{2}-\dfrac{5(x-2)}{6}=\dfrac{x}{3}+11 $

e/ $ x-\dfrac{2x-5}{5}+\dfrac{x+8}{8}=7+\dfrac{x-1}{3} $

f/ $ \dfrac{5x+2}{6}-\dfrac{8x-1}{3}= \dfrac{4x+2}{5}-5 $

Bài 3: Giải các hệ phương trình sau:

a/ $ \dfrac{x-2}{3}+\dfrac{x-2}{4}-\dfrac{x-2}{5}-\dfrac{x-2}{6}=0 $

b/ $ \dfrac{x-1}{2}+\dfrac{x-1}{3}+\dfrac{x-3}{4}= 6 $

c/ $ \dfrac{x-10}{1994}+\dfrac{x-8}{1996}+\dfrac{x-6}{1998}+\dfrac{x-4}{2000}+\dfrac{x-2}{2002} = 5 $

d/ $ \dfrac{x-85}{15}+\dfrac{x-74}{13}+\dfrac{x-67}{11}+\dfrac{x-64}{9} = 10 $
e/ $ \dfrac{x-2002}{5}+\dfrac{x-1992}{10}+\dfrac{x-1982}{15}+\dfrac{x-1972}{20} + \dfrac{x-1962}{25}= 10 $

f/ $ \dfrac{x+50}{15}+\dfrac{x+31}{17}+\dfrac{x+8}{19}+ \dfrac{x-19}{21}+\dfrac{x-50}{23}= – 15 $