Tim x , y, z , biet
\(\frac{x}{3}\)=\(\frac{y}{4}\); \(\frac{y}{3}\)=\(\frac{Z}{5}\) Va 2x - 3y + z=6
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Tim x , y, z , biet
\(\frac{x}{3}\)=\(\frac{y}{4}\); \(\frac{y}{3}\)=\(\frac{Z}{5}\) Va 2x - 3y + z=6
Ta có :\(\frac{3}{x}+\frac{4}{y}+\frac{5}{z}=6\)
\(\Leftrightarrow\frac{6}{2x}+\frac{12}{3y}+\frac{20}{4z}=6\)
\(\Leftrightarrow\frac{6}{2x}+\frac{12}{2x}+\frac{20}{2x}=6\)
\(\Leftrightarrow\frac{6+12+20}{2x}=6\)
\(\Leftrightarrow\frac{19}{x}=6\)
\(\Leftrightarrow x=\frac{19}{6}\)
\(\Leftrightarrow\frac{2}{3}x=\frac{2}{3}.\frac{19}{6}=\frac{19}{9}=y\)
\(\Leftrightarrow\frac{3}{4}y=\frac{3}{4}.\frac{19}{9}=\frac{19}{12}=z\)
Vậy \(\hept{\begin{cases}x=\frac{19}{6}\\y=\frac{19}{9}\\z=\frac{19}{12}\end{cases}}\)
\(\frac{z}{4}=\frac{y}{3}\Rightarrow\frac{z}{20}=\frac{y}{15}^{\left(1\right)}\)
\(\frac{x}{2}=\frac{z}{5}\Rightarrow\frac{x}{8}=\frac{z}{20}^{\left(2\right)}\)
\(\left(1\right),\left(2\right)\Rightarrow\frac{x}{8}=\frac{z}{20}=\frac{y}{15}\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\frac{x}{8}=\frac{z}{20}=\frac{y}{15}=\frac{x+y+z}{8+20+15}=\frac{51}{43}\)
..... ( tới bước này bạn tự làm tiếp nhá )
\(\frac{z}{4}=\frac{y}{3}\Rightarrow\frac{z}{20}=\frac{y}{15}\)
\(\frac{x}{2}=\frac{z}{5}\Rightarrow\frac{x}{8}=\frac{z}{20}\)
Suy ra: \(\frac{x}{8}=\frac{y}{15}=\frac{z}{20}=\frac{x+y+z}{8+15+20}=\frac{51}{43}\)
Vậy \(x=8.\frac{51}{43}=\frac{408}{43}\)
\(y=15.\frac{51}{43}=\frac{765}{43}\)
\(z=20.\frac{51}{43}=\frac{1020}{43}\)
\(\frac{x-5}{3}=\frac{y-4}{4}=\frac{z-3}{5}=\frac{x-5+y-4+z-3}{3+4+5}=\frac{36-12}{12}=\frac{24}{12}=2\)
\(\Rightarrow\hept{\begin{cases}x-5=6\\y-4=8\\z-3=10\end{cases}}\Rightarrow\hept{\begin{cases}x=11\\y=12\\z=13\end{cases}}\)
\(\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{y+z+1+x+z+2+x+y-3}{x+y+z}=2\)
Suy ra
\(x+y+z=\frac{1}{2}\)(1)
\(y+z+1=2x\)(2)
\(x+z+2=2y\)(3)
\(x+y-3=2z\)(4)
(2)-(1) ta có
\(1-x=2x-\frac{1}{2}\Rightarrow3x=\frac{3}{2}\Rightarrow x=\frac{1}{2}\)
\(x+y+z=\frac{1}{2}\Rightarrow y+z=\frac{1}{2}-x\Leftrightarrow y+z=\frac{1}{2}-\frac{1}{2}=0\)
\(y=-z\)
\(x+z+2=\frac{1}{2}+2-y==\frac{5}{2}-y\)
\(\frac{\frac{5}{2}-y}{y}=\frac{5}{2y}-1=2\Leftrightarrow\frac{5}{2y}=3\Leftrightarrow y=\frac{5}{6}\)
\(z=-\frac{5}{6}\)
Ta có: \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\Rightarrow\frac{x^2}{4}=\frac{y^2}{9}=\frac{2z^2}{32}\) và x2 - y2 + 2x2 = 108
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(\frac{x^2}{4}=\frac{y^2}{9}=\frac{2z^2}{32}=\frac{x^2-y^2+2z^2}{4-9+32}=\frac{108}{27}=4\)
\(\Rightarrow\frac{x^2}{4}=4\Rightarrow x=4\)
\(\Rightarrow\frac{y^2}{9}=4\Rightarrow y=6\)
\(\Rightarrow\frac{2z^2}{32}=4\Rightarrow z=8\)
theo dãy tỉ số bằng nhau ta có :
\(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\)<=> \(\frac{x^2}{4}=\frac{y^2}{9}=\frac{2z^2}{32}\)<=> \(\frac{x^2-y^2+2z^2}{4+9+32}=\frac{108}{45}=\frac{12}{5}\)
=> x=245
y=36/5
z= 48/5
\(\frac{x}{2}-2=\frac{y}{3}-2=\frac{z}{4}-2\)
\(\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{4}=\frac{x+y+z}{2+3+4}=\frac{27}{9}=3\)
\(\Rightarrow x=6,y=9,z=12\)
Áp dụng tính chất dãy tỉ số bằng nhau ,ta có:
\(\frac{x-4}{2}=\frac{y-6}{3}=\frac{z-8}{4}=\frac{x+y+z-18}{2+3+4}=1\)
Ta có:\(\frac{x-4}{2}=1\Rightarrow x=6\)
\(\frac{y-6}{3}=1\Rightarrow y=9\)
\(\frac{z-8}{4}=1\Rightarrow z=12\)
Ta có:\(\frac{x}{3}=\frac{y}{4}\)\(\Rightarrow\frac{x}{9}=\frac{y}{12}\)
\(\frac{y}{3}=\frac{z}{5}\)\(\Rightarrow\frac{y}{12}=\frac{z}{20}\)
Suy ra:\(\frac{x}{9}=\frac{y}{12}=\frac{z}{20}\)
Đặt\(\frac{x}{9}=\frac{y}{12}=\frac{z}{20}=k\)
\(\Rightarrow\hept{\begin{cases}x=9k\\y=12k\\z=20k\end{cases}}\)
Mà\(2x-3y+z=6\)
\(\Rightarrow2.9k-3.12k+20k=6\)
\(\Leftrightarrow18k-36k+20k=6\)
\(\Leftrightarrow2k=6\)
\(\Leftrightarrow k=3\)
\(\Rightarrow\hept{\begin{cases}x=3.9=27\\y=3.12=36\\z=3.20=60\end{cases}}\)(Thỏa mãn)
Vậy\(\hept{\begin{cases}x=27\\y=36\\z=60\end{cases}}\)
Linz
Ta có : \(\hept{\begin{cases}\frac{x}{3}=\frac{y}{4}\\\frac{y}{3}=\frac{z}{5}\end{cases}}\Rightarrow\hept{\begin{cases}\frac{x}{9}=\frac{y}{12}\\\frac{y}{12}=\frac{z}{20}\end{cases}\Rightarrow\frac{x}{9}=\frac{y}{12}=\frac{z}{20}}\)
=> \(\frac{2x}{18}=\frac{3y}{36}=\frac{z}{20}=\frac{2x-3y+z}{18-36+20}=\frac{6}{2}=3\)(dãy tỉ số bằng nhau)
=> x = 27 ; y = 36 ; z = 60