\(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\) và x . y = 1200
Tìm 3 số x, y, z biết \(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\) và x.y = 1200.
Tìm x,y,z biết:
\(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\) và x.y=1200
\(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\)
\(\Rightarrow\dfrac{x-9}{15}=\dfrac{y-12}{20}=\dfrac{z-24}{40}=k\)
\(\Rightarrow\left(15k+9\right)\left(20k+12\right)=1200\)
\(\Rightarrow3.4\left(5k+3\right)\left(5k+3\right)=1200\)
\(\Rightarrow\left(5k+3\right)\left(5k+3\right)=1200:3:4\)
\(\Rightarrow\left(5k+3\right)^2=100\)
\(\Rightarrow\left[{}\begin{matrix}5k+3=10\\5k+3=-10\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}k=\dfrac{7}{5}\\k=\dfrac{-13}{5}\end{matrix}\right.\)
+) Với \(k=\dfrac{7}{5}\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{7}{5}.15+9\\y=\dfrac{7}{5}.20+12\\z=\dfrac{7}{5}.40+24\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=30\\y=40\\z=80\end{matrix}\right.\)
+) Với \(k=\dfrac{-13}{5}\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{-13}{5}.15+9\\y=\dfrac{-13}{5}.20+12\\z=\dfrac{-13}{5}.40+24\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=-30\\y=-40\\z=-80\end{matrix}\right.\)
Vậy ................................
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Tìm x , y sao cho \(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\) và xy = 1200
\(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\)
\(\Rightarrow\dfrac{x-9}{15}=\dfrac{y-12}{20}=\dfrac{z-24}{40}=k\)
\(\Rightarrow\left(15k+9\right)\left(20k+12\right)=1200\)
\(\Rightarrow3.4\left(5k+3\right)\left(5k+3\right)=1200\)
\(\Rightarrow\left(5k+3\right)\left(5k+3\right)=1200:3:4\)
\(\Rightarrow\left(5k+3\right)^2=100\)
\(\Rightarrow\left[{}\begin{matrix}5k+3=10\\5k+3=-10\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}k=\dfrac{7}{5}\\k=-\dfrac{13}{5}\end{matrix}\right.\)
+) Với \(k=\dfrac{7}{5}\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{7}{5}.15+9\\y=\dfrac{7}{5}.20+12\\z=\dfrac{7}{5}.40+24\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=30\\y=40\\z=80\end{matrix}\right.\)
+) Với \(k=-\dfrac{13}{5}\)
\(\Rightarrow\left\{{}\begin{matrix}x=-\dfrac{13}{5}.15+9\\y=-\dfrac{13}{5}.20+12\\z=-\dfrac{13}{5}.40+24\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-30\\y=-40\\z=-80\end{matrix}\right.\)
Vậy...................................
Ta có: (dfrac{15}{x-9}=dfrac{20}{y-12}=dfrac{40}{z-24})
(Rightarrowdfrac{x-9}{15}=dfrac{y-12}{20}=dfrac{z-24}{40})
(Rightarrowdfrac{x}{15}-dfrac{3}{5}=dfrac{y}{20}-dfrac{3}{5}=dfrac{z}{40}-dfrac{3}{5})
(Rightarrowdfrac{x}{15}=dfrac{y}{20}=dfrac{z}{40})
Đặt (dfrac{x}{15}=dfrac{y}{20}=dfrac{z}{40}=k)
(Rightarrowleft[{}egin{matrix}x=15k\y=20k\z=40kend{matrix} ight.) (left(1 ight))
Thay (left(1 ight)) vào đề bài:
(15k.20k=1200)
(Rightarrow300k^2=1200)
(Rightarrow k^2=4Rightarrow k=pm2)
Khi (k=2) thì: (left[{}egin{matrix}x=15.2=30\y=20.2=40\z=40.2=80end{matrix} ight.)
Khi (k=-2) (thì:) (left[{}egin{matrix}x=15left(-2 ight)=-30\y=20left(-2 ight)=-40\z=40left(-2 ight)=-80end{matrix} ight.)
Vậy ta tìm đc 2 cặp giá trị (x,y,z) thỏa mãn đề bài:
(left[{}egin{matrix}x=30\y=40\z=80end{matrix} ight.) và (left[{}egin{matrix}x=-30\y=-40\z=-80end{matrix} ight.).
Tìm x,y,z:
a/\(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\)và x.y=1200
b/15x=-10y=6z và xyz=30000
\(a.\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\&xy=1200\)
\(\Leftrightarrow\dfrac{15}{20}=\dfrac{x-9}{y-12}\Leftrightarrow\dfrac{3}{4}=\dfrac{x-9}{y-12}\)
\(\Rightarrow\dfrac{9}{12}=\dfrac{x-9}{y-12}\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\dfrac{9}{12}=\dfrac{x-9}{y-12}=\dfrac{x-9+9}{y-12+12}=\dfrac{x}{y}\)
\(\Rightarrow\dfrac{x}{y}=\dfrac{xy}{y^2}=\dfrac{x^2}{xy}\)
Từ \(\dfrac{3}{4}=\dfrac{xy}{y^{^2}}\Rightarrow\dfrac{3}{4}=\dfrac{1200}{y^2}\Rightarrow y^2=1200.\dfrac{4}{3}=1600\)
\(\Rightarrow y=\sqrt{1600}=\pm40\)
+ TH1: \(y=40\Rightarrow x=30\)
\(\dfrac{15}{x-9}=\dfrac{40}{z-24}\Rightarrow z=80\) (tự giải pt)
+ TH2: \(y=-40\Rightarrow x=-30\)
\(\dfrac{15}{x-9}=\dfrac{40}{z-4}\Rightarrow z=-80\) (tự giải pt)
Vậy, các cặp \(\left(x;y;z\right)\) thỏa mãn là \(\left(30;40;80\right)\&\left(-30;-40;-80\right)\)
\(b.15x=-10y=6z\&xyz=30000\)
\(\Rightarrow\left\{{}\begin{matrix}15x=-10y\\-10y=6z\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{-10}=\dfrac{y}{15}\\\dfrac{y}{6}=\dfrac{z}{-10}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\dfrac{x}{-20}=\dfrac{y}{30}\\\dfrac{y}{30}=\dfrac{z}{-50}\end{matrix}\right.\Rightarrow\dfrac{x}{-20}=\dfrac{y}{30}=\dfrac{z}{-50}\)
Đặt \(\dfrac{x}{-20}=\dfrac{y}{30}=\dfrac{z}{-50}=k\Rightarrow x=-20k;y=30k;z=-50k\)
\(\Rightarrow xyz=30000\Rightarrow-20k.30k.\left(-50k\right)=30000\Rightarrow30000k^3=30000\)
\(\Rightarrow k^3=1\Rightarrow k=1\)
\(\Rightarrow\left\{{}\begin{matrix}x=-20\\y=30\\z=-50\end{matrix}\right.\)
Tìm 3 số x, y, z biết \(\dfrac{x}{15}\)=\(\dfrac{y}{20}\)=\(\dfrac{z}{40}\) và x.y = 1200.
Đặt \(\dfrac{x}{15}=\dfrac{y}{20}=\dfrac{z}{40}=k\Leftrightarrow x=15k;y=20k;z=40k\)
\(xy=1200\\ \Leftrightarrow300k^2=1200\\ \Leftrightarrow k^2=4\Leftrightarrow\left[{}\begin{matrix}k=2\\k=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=30;y=40;z=80\\x=-30;y=-40;z=-80\end{matrix}\right.\)
Tìm x,y,z trong dãy tỉ số bằng nhau
1)\(\dfrac{3x}{8}=\dfrac{3y}{64}=\dfrac{3z}{216}\)và \(2x^2+2y^2.z^2=1\)
2) \(\dfrac{2x+1}{5}=\dfrac{4y-5}{9}=\dfrac{2x+4y-4}{7x}\)
3) \(\dfrac{x^3+y^3}{6}=\dfrac{x^3-2y^3}{4}\)và x6 . y6 =14
4) \(\dfrac{x+4}{6}=\dfrac{3y-1}{8}=\dfrac{3y-x-5}{x}\)
5) \(\dfrac{3}{x-1}=\dfrac{4}{y-2}=\dfrac{5}{z-3}\)và x.y.z=192
6)\(\dfrac{x-y}{3}=\dfrac{x+y}{13}=\dfrac{x.y}{200}\)
7)\(\dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z+2}{4}=\dfrac{x+y+z+2}{2x+5}\)
8) \(\dfrac{15}{x-9}=\dfrac{20}{y-12}=\dfrac{40}{z-24}\)và x.y = 1200
9)\(\dfrac{40}{x-30}=\dfrac{20}{y-15}=\dfrac{28}{z-21}\) và x.y.z = 22400
10)15x = -10y =6z và x.y.z = -30000
11) Cho\(\dfrac{x+1}{3}=\dfrac{y-2}{5}=\dfrac{2z+14}{9}\)và x+z=y
12) Cho \(\dfrac{x}{3}=\dfrac{y}{4}\)và \(\dfrac{y}{5}=\dfrac{z}{6}\).Tính M=\(\dfrac{2x+3y+4z}{3x+4y+5z}\)
tìm các số x,y,z, biết: 15/x-9=20/y-12=40/z-24 và xy=1200
4) \(\dfrac{x}{0,5}=\dfrac{y}{0,3}=\dfrac{z}{0,2}\) và -y+x=1
6) \(\dfrac{x+11}{13}=\dfrac{y+12}{14}=\dfrac{z+13}{15}\)và x+y+z=6
7) 5x=4y và x.y=20
7) 5x=4y ⇒\(\dfrac{x}{4}=\dfrac{y}{5}\)
Nhân cả hai vế với \(\dfrac{x}{4}\), ta có: \(\left(\dfrac{x}{4}\right)^2=\dfrac{x}{4}.\dfrac{y}{5}=\dfrac{xy}{20}=\dfrac{20}{20}=1\)
\(\left(\dfrac{x}{4}\right)^2=1\Rightarrow\left[{}\begin{matrix}\dfrac{x}{4}=1\\\dfrac{x}{4}=-1\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=4\\x=-4\end{matrix}\right.\)
⇒ \(\left[{}\begin{matrix}y=5\\y=-5\end{matrix}\right.\)
4) áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x}{0,5}=\dfrac{y}{0,3}=\dfrac{z}{0,2}=\dfrac{z-y+x}{0,2-0,3+0,5}=\dfrac{1}{\dfrac{2}{5}}=\dfrac{5}{2}\)
\(\dfrac{x}{0,5}=\dfrac{5}{2}\Rightarrow x=\dfrac{5}{4}\)
\(\dfrac{y}{0,3}=\dfrac{5}{2}\Rightarrow y=\dfrac{3}{4}\)
\(\dfrac{z}{0,2}=\dfrac{5}{2}\Rightarrow z=\dfrac{1}{2}\)
6) áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x+11}{13}=\dfrac{y+12}{14}=\dfrac{z+13}{15}=\dfrac{x+11+y+12+z+13}{13+14+15}=\dfrac{42}{42}=1\)
\(\dfrac{x+11}{13}=1\Rightarrow x=2\)
\(\dfrac{y+12}{13}=1\Rightarrow y=1\)
\(\dfrac{z+13}{15}=1\Rightarrow z=2\)
7) \(5x=4y\Rightarrow\dfrac{x}{4}=\dfrac{y}{5}=k\)
\(\Rightarrow x=4k,y=5k\)
\(x.y=20\\ \Rightarrow4k.5k=20\\ \Rightarrow20k^2=20\\ \Rightarrow k^2=1\\ \Rightarrow\left[{}\begin{matrix}k=-1\\k=1\end{matrix}\right.\)
\(x=4k\Rightarrow\left[{}\begin{matrix}x=-4\\x=4\end{matrix}\right.\)
\(y=5k\Rightarrow\left[{}\begin{matrix}y=-5\\y=5\end{matrix}\right.\)
Vậy \(\left(x,y\right)=\left\{\left(-4;-5\right);\left(4;5\right)\right\}\)
15/x-9=20/y-12=40/z-24 va x*y= 1200