Bài 1: 

Ta có: \(3x=2y\)

nên \(\dfrac{x}{2}=\dfrac{y}{3}\)

mà x+y=-15

nên Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{x+y}{2+3}=\dfrac{-15}{5}=-3\)

Do đó:

\(\left\{{}\begin{matrix}\dfrac{x}{2}=-3\\\dfrac{y}{3}=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-6\\y=-9\end{matrix}\right.\)

Vậy: (x,y)=(-6;-9)

Bài 2: 

a) Ta có: \(\dfrac{x}{4}=\dfrac{y}{3}=\dfrac{z}{5}\)

mà x+y-z=20

nên Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{x}{4}=\dfrac{y}{3}=\dfrac{z}{5}=\dfrac{x+y-z}{4+3-5}=\dfrac{20}{2}=10\)

Do đó:

\(\left\{{}\begin{matrix}\dfrac{x}{4}=10\\\dfrac{y}{3}=10\\\dfrac{z}{5}=10\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=40\\y=30\\z=50\end{matrix}\right.\)

Vậy: (x,y,z)=(40;30;50)

Bài 2: 

b) Ta có: \(\dfrac{y}{3}=\dfrac{z}{7}\)

nên \(\dfrac{y}{12}=\dfrac{z}{28}\)

mà \(\dfrac{x}{11}=\dfrac{y}{12}\)

nên \(\dfrac{x}{11}=\dfrac{y}{12}=\dfrac{z}{28}\)

hay \(\dfrac{2x}{22}=\dfrac{y}{12}=\dfrac{z}{28}\)

mà 2x-y+z=152

nên Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{2x}{22}=\dfrac{y}{12}=\dfrac{z}{28}=\dfrac{2x-y+z}{22-12+28}=\dfrac{152}{38}=4\)

Do đó:

\(\left\{{}\begin{matrix}\dfrac{x}{11}=4\\\dfrac{y}{12}=4\\\dfrac{z}{28}=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=44\\y=48\\z=112\end{matrix}\right.\)

Vậy: (x,y,z)=(44;48;112)

gợi ý nè:

thử cộng chúng lại xem

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

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

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

\(x+y+z\) = \(\dfrac{x+y+z}{2.\left(x+y+z\right)}\) = \(\dfrac{1}{2}\) (1)

\(\dfrac{x}{y+z+1}\) = \(\dfrac{1}{2}\) ⇒ 2\(x\) = y+z+1 

⇒ 2\(x\) + \(x\) = \(x+y+z+1\) (2)

 Thay (1) vào (2) ta có: 2\(x\) + \(x\) = \(\dfrac{1}{2}\) + 1

                                      3\(x\)      = \(\dfrac{3}{2}\) ⇒ \(x=\dfrac{1}{2}\)

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

Thay (1) vào (3) ta có: 2y + y = \(\dfrac{1}{2}\) + 2 

                                   3y = \(\dfrac{5}{2}\) ⇒ y = \(\dfrac{5}{6}\)

Thay \(x=\dfrac{1}{2};y=\dfrac{5}{6}\) vào (1) ta có: \(\dfrac{1}{2}+\dfrac{5}{6}+z\) = \(\dfrac{1}{2}\)

                                                              \(\dfrac{5}{6}\) + z = 0 ⇒ z = - \(\dfrac{5}{6}\)

Kết luận: (\(x;y;z\)) = (\(\dfrac{1}{2}\); \(\dfrac{5}{6}\); - \(\dfrac{5}{6}\))

TH1: x + y + z $\ne$ 0

Áp dụng tính chất dãy tỉ số bằng nhau, ta có:

$\frac{x}{y+z+1}$ = $\frac{y}{x+z+2}$ = $\frac{z}{x+y-3}$ = $\frac{x+y+z}{y+z+1+x+z+2+x+y-3}$ 

              = $\frac{x+y+z}{x+y+z+x+y+z}$ = $\frac{x+y+z}{2(x+y+z)}$ = $\frac{1}{2}$ 

⇒ x + y + z = $\frac{1}{2}$

⇒ x + y       = $\frac{1}{2}$ - z

    x + z        = $\frac{1}{2}$ - y

    y + z        = $\frac{1}{2}$ - x

Thay y + z + 1 = $\frac{1}{2}$ - x + 1

⇒ $\frac{x}{\frac{1}{2}-x+1}$ = $\frac{1}{2}$

⇒ 2x = $\frac{1}{2}$ - x + 1

⇒ 2x + x = $\frac{1}{2}$ + 1

⇒  3x   =  $\frac{3}{2}$

⇒   x    = $\frac{1}{2}$

Thay x + z + 2 = $\frac{1}{2}$ - y + 2

⇒ $\frac{y}{\frac{1}{2}-y+2}$ = $\frac{1}{2}$

⇒ 2y = $\frac{1}{2}$ - y + 2

⇒ 2y + y = $\frac{1}{2}$ + 2

⇒   3y  = $\frac{5}{2}$

⇒     y   = $\frac{5}{6}$

Thay x + y - 3 = $\frac{1}{2}$ - z - 3

⇒ $\frac{z}{\frac{1}{2}-z-3}=$\frac{1}{2}$

⇒ 2z = $\frac{1}{2}$ - z - 3

⇒ 2z + z = $\frac{1}{2}$ - 3

⇒  3z  = $\frac{-5}{2}$

⇒   z   = $\frac{-5}{6}$

TH2: x + y + z = 0

⇒ $\frac{x}{y+z+1}$ = $\frac{y}{x+z+2}$ = $\frac{z}{x+y-3}$ = 0

⇒ x = y = z = 0

https://olm.vn/cau-hoi/tim-tat-ca-cac-so-xyz-biet-dfracxyz1dfracyxz2dfraczxy-3xyz-giair-chi-tiet-ho-e-vs-a.8297156371934

a) Với \(x+y+z=0\) ta tìm được \(\left(x;y;z\right)\rightarrow\left(0;0;0\right)\)

Với \(x+y+z\ne0\) áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{x}{y+z+1}=\dfrac{y}{z+x+1}=\dfrac{z}{x+y-2}=\dfrac{x+y+z}{2\left(x+y+z\right)}=\dfrac{1}{2}\)

Hay: \(x+y+z=\dfrac{1}{2}\Leftrightarrow\left\{{}\begin{matrix}y+z=\dfrac{1}{2}-x\\x+z=\dfrac{1}{2}-y\\x+y=\dfrac{1}{2}-z\end{matrix}\right.\)

Thay vào đề bài ta được:

\(\dfrac{x}{\dfrac{1}{2}-x+1}=\dfrac{y}{\dfrac{1}{2}-y+1}=\dfrac{z}{\dfrac{1}{2}-z-2}=\dfrac{1}{2}\) Dễ dàng tìm được x;y;z

b) Theo đề bài ta có sẵn x+y+z khác 0

Áp dụng dãy tỉ số rồi làm tương tự câu a

Áp dụng t/c dtsbn ta có:

\(\dfrac{y+z+1}{x}=\dfrac{x+z+2}{y}=\dfrac{x+y-3}{z}=\dfrac{1}{x+y+z}=\dfrac{y+z+1+x+z+2+x+y-3}{x+y+z}=\dfrac{2\left(x+y+z\right)}{x+y+z}=2\)

\(\dfrac{1}{x+y+z}=2\Rightarrow2x+2y+2z=1\Rightarrow x+y+z=0,5\Rightarrow\left\{{}\begin{matrix}x+y=0,5-z\\y+z=0,5-x\\x+z=0,5-y\end{matrix}\right.\\ \dfrac{y+z+1}{x}=2\Rightarrow y+z+1=2x\Rightarrow0,5-x+1=2x\Rightarrow x=0,5\\ \dfrac{x+z+2}{y}=2\Rightarrow x+z+2=2y\Rightarrow0,5-y+2=2y\Rightarrow y=\dfrac{5}{6}\\ \dfrac{x+y-3}{z}=2\Rightarrow x+y-3=2z\Rightarrow0,5-z-3=2z\Rightarrow z=-\dfrac{5}{6}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x}{y+z+1}=\dfrac{y}{x+z+1}=\dfrac{z}{x+y-2}=x+y+z=\dfrac{x+y+z}{y+z+1+x+z+1+x+y-2}=\dfrac{x+y+z}{2x+2y+2z}=\dfrac{x+y+z}{2\left(x+y+z\right)}=\dfrac{1}{2}\)

\(\dfrac{x}{y+z+1}=\dfrac{1}{2}\Rightarrow y+z+1=2x\Rightarrow y+z=2x-1\left(1\right)\)

\(\dfrac{y}{x+z+1}=\dfrac{1}{2}\Rightarrow x+z+1=2y\Rightarrow x+z=2y-1\left(2\right)\)

\(\dfrac{z}{x+y-2}=\dfrac{1}{2}\Rightarrow x+y-2=2z\)

\(x+y+z=\dfrac{1}{2}\left(3\right)\)

Thay (1) vào (3) ta có:

\(x+y+z=\dfrac{1}{2}\\ \Rightarrow x+2x-1=\dfrac{1}{2}\\ \Rightarrow3x=\dfrac{3}{2}\\ \Rightarrow x=\dfrac{1}{2}\)

Thay (2) vào (3) ta có:

\(x+y+z=\dfrac{1}{2}\\ \Rightarrow y+2y-1=\dfrac{1}{2}\\ \Rightarrow3y=\dfrac{3}{2}\\ \Rightarrow y=\dfrac{1}{2}\)

Ta có:

\(x+y+z=\dfrac{1}{2}\\ \Rightarrow\dfrac{1}{2}+\dfrac{1}{2}+z=\dfrac{1}{2}\\ \Rightarrow z=-\dfrac{1}{2}\)

TH1: \(x+y+z=0\Rightarrow x=y=z=0\)

TH2: \(x+y+z\ne0\)

\(x+y+z=\dfrac{x}{y+z+1}=\dfrac{y}{x+z+1}=\dfrac{z}{x+y-2}=\dfrac{x+y+z}{2\left(x+y+z\right)}=\dfrac{1}{2}\)

\(\Leftrightarrow\)\(\left\{{}\begin{matrix}2x+2y+2z=1\\2x=y+z+1\\2y=x+z+1\\2z=x+y-2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}2x+2y+2z=1\\2x+2y+2z=3y+3z+1\\2x+2y+2z=3x+3z+1\\2x+2y+2z=3x+3y-2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x+2y+2z=1\\y+z=0\\x+z=0\\x+y=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2.1+2z=1\\y=-z\\x=-z\\x+y=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}z=-\dfrac{1}{2}\\x=\dfrac{1}{2}\\y=\dfrac{1}{2}\\\end{matrix}\right.\)

Vậy \(\left(x;y;z\right)=\left(0;0;0\right);\left(\dfrac{1}{2};\dfrac{1}{2};-\dfrac{1}{2}\right)\)

\(\Rightarrow\dfrac{z+y+1}{x}=\dfrac{x+z+1}{y}=\dfrac{x+y-2}{z}=\dfrac{2\left(x+y+z\right)}{x+y+z}=2=x+y+z\\ \Rightarrow\left\{{}\begin{matrix}z+y+1=2x\\x+z+1=2y\\x+y-2=2z\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}y+z=2x-1\\x+z=2y-1\\x+y=2z+2\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}2x-1=2-x\\2y-1=2-y\\2z+2=2-z\end{matrix}\right.\Rightarrow\left(x,y,z\right)=\left(1;1;0\right)\)