a,\(\dfrac{6}{2x+1}=\dfrac{2}{7}\)

⇒\(\dfrac{6}{2x+1}=\dfrac{6}{21}\)

⇒\(2x+1=21\)

\(2x=21-1\)

\(2x=20\)

⇒\(x=10\)

a, \(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{9900}\)

\(\Rightarrow\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\)

\(\Rightarrow\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{100}\)

\(\Rightarrow\dfrac{1}{1}-\dfrac{1}{100}\)

\(\Rightarrow\dfrac{99}{100}\)

`[-1]/2+[-1]/6+[-1]/12+[-1]/20+[-1]/30+[-1]/42+[-1]/56+[-1]/72+[-1]/90`

`=(-1)(1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90)`

`=(-1)(1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10)`

`=(-1)(1-1/10)`

`=(-1). 9/10=-9/10`

A = \(\dfrac{-1}{2}\) + \(\dfrac{-1}{6}\)+ \(\dfrac{-1}{12}\)+ \(\dfrac{-1}{20}\)+ \(\dfrac{-1}{30}\)+ \(\dfrac{-1}{42}\)+ \(\dfrac{-1}{56}\)+ \(\dfrac{-1}{72}\)+ \(\dfrac{-1}{90}\)

A = \(\dfrac{-1}{2}\) + \(\dfrac{-1}{2\times3}\)+ \(\dfrac{-1}{3\times4}\)+ \(\dfrac{-1}{4\times5}\)+ \(\dfrac{-1}{5\times6}\)+ \(\dfrac{-1}{6\times7}\)+ \(\dfrac{-1}{7\times8}\)+ \(\dfrac{-1}{8\times9}\)+ + \(\dfrac{-1}{9\times10}\)

A = - (\(\dfrac{1}{2}\)+ \(\dfrac{1}{2}\)-\(\dfrac{1}{3}\)+ \(\dfrac{1}{3}\)-\(\dfrac{1}{4}\)+ \(\dfrac{1}{4}\)- \(\dfrac{1}{5}\)+ \(\dfrac{1}{5}\)-\(\dfrac{1}{6}\)+ \(\dfrac{1}{6}\)-\(\dfrac{1}{7}\)+\(\dfrac{1}{7}\)-\(\dfrac{1}{8}\)+\(\dfrac{1}{8}\)-\(\dfrac{1}{9}\)+ \(\dfrac{1}{9}\)-\(\dfrac{1}{10}\))

A = -(1-\(\dfrac{1}{10}\))

A = \(\dfrac{-9}{10}\)

` a/` 

` 3 1/5 : 2 1/3 : y = 12/7 `

` 48/35 : y = 12/7 `

` y = 48/35 : 12/7 `

` y = 48/35 xx 7/12 `

` y = 4/5`

Vậy ` y = 4/5` 

`b/`

` 3 : y xx 3 1/2 = 2/3 xx 3/4 `

` 3 : y xx 7/2 =  1/2`

` 3 : y = 1/2 : 7/2 `

` 3 : y = 1/2 xx 2/7 `

` 3 : y = 1/7 `

`      y = 3 : 1/7 `

`      y = 3 xx 7`

`      y = 21 `

 Vậy ` y = 21 `

`c/` 

` 3 2/3 - y + 1 3/4 = 2`

` 11/3 - y + 7/4 = 2 `

`   11/3 -y  = 2 - 7/4`

`    11/3 - y = 1/4 `

`        y = 11/3 - 1/4 `

`        y = 41/12  `

Vậy ` y = 41/12`

=9/10-(1/2+1/6+...+1/90)

=9/10-(1-1/2+1/2-1/3+...+1/9-1/10)

=9/10-9/10=0

sau đây là phần chữa của mình: 

\(=\dfrac{1}{2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{9.10}\)

\(=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{9}-\dfrac{1}{10}\)

 \(=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{10}\)

= \(\dfrac{3}{10}\)

= \(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{9.10}\)

\(=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{9}-\dfrac{1}{10}\)

= \(\dfrac{1}{2}-\dfrac{1}{10}\)

= \(\dfrac{2}{5}\)

chỗ đầu mình quên cộng thêm 1/2 rồi bạn nhớ bổ sung nha thì sẽ ra kết quả đúng nhé

b:

ĐKXĐ: x<>0

 \(\dfrac{2}{x}+\dfrac{y}{3}=\dfrac{1}{6}\)

=>\(\dfrac{6+xy}{3x}=\dfrac{1}{6}\)

=>\(6\left(6+xy\right)=3x\)

=>\(x=2\left(6+xy\right)=12+2xy\)

=>\(x\left(1-2y\right)=12\)

mà x,y là các số nguyên

nên \(\left(x;1-2y\right)\in\left\{\left(12;1\right);\left(-12;-1\right);\left(4;3\right);\left(-4;-3\right)\right\}\)

=>\(\left(x,y\right)\in\left\{\left(12;0\right);\left(-12;1\right);\left(4;-1\right);\left(-4;2\right)\right\}\)

c: ĐKXĐ: y<>-1

\(\dfrac{x}{3}+\dfrac{1}{y+1}=\dfrac{1}{6}\)

=>\(\dfrac{xy+x+3}{3\left(y+1\right)}=\dfrac{1}{6}\)

=>\(\dfrac{2\left(xy+x+3\right)}{6\left(y+1\right)}=\dfrac{y+1}{6\left(y+1\right)}\)

=>\(2xy+2x+6=y+1\)

=>\(2x\left(y+1\right)-\left(y+1\right)=-6\)

=>\(\left(2x-1\right)\left(y+1\right)=-6\)

mà x,y là các số nguyên

nên \(\left(2x-1;y+1\right)\in\left\{\left(1;-6\right);\left(-1;6\right);\left(3;-2\right);\left(-3;2\right)\right\}\)

=>\(\left(x,y\right)\in\left\{\left(1;-7\right);\left(0;5\right);\left(2;-3\right);\left(-1;1\right)\right\}\)

y.(1/2+1/6+1/12+1/20)=8/5

y.4/5=8/5

y=8/5:4/5

y=8/5.5/4

y=2

\(y\times\dfrac{1}{2}+y\times\dfrac{1}{6}+y\times\dfrac{1}{12}+y\times\dfrac{1}{20}=\dfrac{8}{5}\)

\(y\times\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}\right)=\dfrac{8}{5}\)

\(y\times\dfrac{4}{5}=\dfrac{8}{5}\)

\(\dfrac{8}{5}:\dfrac{4}{5}\)

\(y=\dfrac{8}{4}=2\)

`y(1/2 + 1/6 + 1/12 + 1/20) = 8/5`

`y(1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5) = 8/5`

`4/5y = 8/5`

`=> y = 2`.