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Ta có :\(15x=10y=6z\Rightarrow\hept{\begin{cases}15x=10y\\10y=6z\end{cases}}\Rightarrow\hept{\begin{cases}3x=2y\\5y=3z\end{cases}}\Rightarrow\hept{\begin{cases}\frac{x}{2}=\frac{y}{3}\\\frac{y}{3}=\frac{z}{5}\end{cases}}\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\)
Đặt \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}=k\Rightarrow\hept{\begin{cases}x=2k\\y=3k\\z=5k\end{cases}}\)
Khi đó 5x3 + 2y3 - z3 = 31
=> 5(2k)3 + 2(3k)3 - (5k)3 = 31
=> 40k3 + 54k3 - 125k3 = 31
=> -31k3 = 31
=> k3 = -1
=> k = -1
=> x = -2 ; y = -3 ; z = -5
b) Ta có 7x = 14y = 6z => \(\hept{\begin{cases}7x=14y\\14y=6z\end{cases}}\Rightarrow\hept{\begin{cases}x=2y\\7y=3z\end{cases}}\Rightarrow\hept{\begin{cases}\frac{x}{2}=\frac{y}{1}\\\frac{y}{3}=\frac{z}{7}\end{cases}}\Rightarrow\hept{\begin{cases}\frac{x}{6}=\frac{y}{3}\\\frac{y}{3}=\frac{z}{7}\end{cases}}\Rightarrow\frac{x}{6}=\frac{y}{3}=\frac{z}{7}\)
Đặt \(\frac{x}{6}=\frac{y}{3}=\frac{z}{7}=k\Rightarrow\hept{\begin{cases}x=6k\\y=3k\\z=7k\end{cases}}\)
Khi đó 2x2 - 3y2 = 5
<=> 2.(6k)2 - 3.(3k)2 = 5
=> 72k2 - 27k2 = 5
=> 45k2 = 5
=> k2 = 1/9
=> k = \(\pm\frac{1}{3}\)
Nếu k = 1/3 => x = 2 ; y = 1 ; z = 7/3
Nếu k = -1/3 => x = -2 ; y = - 1 ; z = -7/3
Vậy các cặp (x;y;z) thỏa mãn là : (2;1;7/3) ; (-2 ; - 1; -7/3)
c) Ta có : \(3x=8y=5z\Rightarrow\frac{3x}{120}=\frac{8y}{120}=\frac{5z}{120}\Rightarrow\frac{x}{40}=\frac{y}{15}=\frac{z}{24}\)
Đặt \(\frac{x}{40}=\frac{y}{15}=\frac{z}{24}=k\Rightarrow\hept{\begin{cases}x=40k\\y=15k\\z=24k\end{cases}}\)
Khi đó |x - 2y| = 5
<=> |40k - 2.15k| = 5
=> |10k| = 5
=> \(\orbr{\begin{cases}10k=5\\10k=-5\end{cases}}\Rightarrow\orbr{\begin{cases}k=\frac{1}{2}\\k=-\frac{1}{2}\end{cases}}\)
Nếu k = 5 => x = 20 ; y = 7,5 ; z = 12
Nếu k = -5 => x = -20 ; y =-7,5 ; z = -12
d) 4x = 5y = 6z => \(\frac{4x}{60}=\frac{5y}{60}=\frac{6z}{60}\Rightarrow\frac{x}{15}=\frac{y}{12}=\frac{z}{10}\)
Đặt \(\frac{x}{15}=\frac{y}{12}=\frac{z}{10}=k\Rightarrow\hept{\begin{cases}x=15k\\y=12k\\z=10k\end{cases}}\)
Khi đó (3x - 2y)2 = 16
<=> (3.15k - 2.12k)2 = 16
=> (45k -24k)2 = 16
=> (21k)2 = 16
=> \(\orbr{\begin{cases}21k=4\\21k=-4\end{cases}}\Rightarrow\orbr{\begin{cases}k=\frac{4}{21}\\k=-\frac{4}{21}\end{cases}}\)
Nếu k = 4/21 => x = 20/7 ; y = 16/7 ; z = 40/21
Nếu k = -4/21 => x = -20/7 ; y = -16/7 ; z = -40/21
![](https://rs.olm.vn/images/avt/0.png?1311)
\(15x=-10y\) => \(\frac{x}{-10}=\frac{y}{15}\) => \(\frac{x}{-2}=\frac{y}{3}\)
\(-10y=6z\) => \(\frac{y}{6}=\frac{z}{-10}\) => \(\frac{y}{3}=\frac{z}{-5}\)
=> \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{5}\)
=> \(\left(\frac{x}{2}\right)^3=\left(\frac{y}{-3}\right)^3=\left(\frac{z}{5}\right)^3=\frac{xyz}{2.-3.5}=\frac{-30000}{-30}=1000\)
=> x = 20
y = -30
z = 50
Chúc bạn làm bài tốt
\(15x=-10y=6z\Rightarrow\frac{15x}{30}=\frac{-10y}{30}=\frac{6z}{30}\)
\(\Rightarrow\)\(\frac{x}{2}=\frac{y}{-3}=\frac{z}{5}\)
Đặt \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{5}=n\)
\(\Rightarrow x=2n,y=-3n,z=5n\)
\(\Rightarrow xyz=2n.-3n.5n\)
\(=-30n^3=-30000\Rightarrow n^3=-1000=-10^3\)
\(\Rightarrow n=-10\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(15x=10y=6z\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}\)
Áp dụng t/c dtsbn:
\(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=\dfrac{x+y+z}{2+3+5}=\dfrac{20}{10}=2\)
\(\Rightarrow\left\{{}\begin{matrix}x=2.2=4\\y=2.3=6\\z=2.5=10\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
15x = -10y = 6z
<=> \(\frac{15x}{30}=\frac{-10y}{30}=\frac{6z}{30}\)
<=> \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{5}\)
Đặt \(\frac{x}{2}=\frac{y}{-3}=\frac{z}{5}=k\Rightarrow\hept{\begin{cases}x=2k\\y=-3k\\z=5k\end{cases}}\)
Ta có: xyz = -30000
=> 2k.(-3k).5k = -30000
=> -30k3 = -30000
=> k3 = 1000
=> k = 10
=> x = 20, y = -30, z = 50
Vì 15x = -10y = 6z => \(\frac{15x}{30}=\frac{-10y}{30}=\frac{6z}{30}\) => \(\frac{x}{2}=\frac{-y}{3}=\frac{z}{5}\)
Đặt : \(\frac{x}{2}=\frac{-y}{3}=\frac{z}{5}=k\), ta có : x = 2k ; y = (-3).k ; x = 5k
=> x.y.z = 2 .k. ( -3 ). k.5.k = -30.k3 = -30000
=> k3 = 1000 => k = 10 => x = 10. 2 = 20
=> y = 10. ( - 3 ) = -30
=> z = 10.5 = 50
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\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{1}{2}x-\dfrac{3}{5}>\dfrac{2}{5}\\\dfrac{1}{2}x-\dfrac{3}{5}< -\dfrac{2}{5}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\dfrac{1}{2}x>1\\\dfrac{1}{2}x< \dfrac{1}{5}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x>2\\x< \dfrac{2}{5}\end{matrix}\right.\)
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Ta có : 15x = 6z
=> x = 6/15z
-10y = 6z
=> y= -3/5z
=> xyz = -30000
<=> (6/15z) . (-3/5z) . z = -30000
<=> z^3 .( -6/25) = -30000
<=> z^3 = 125000
<=> z = 50
=> y = -30
=> x = 20
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có :
15x = -10y
=> 3.x = -2.y => x/-2 = y/3 [1]
-10y = 6.z
=> -5.y = 3.z => y/3 = z/-5 [2]
Từ [1] và [2] => x/-2 = y/3 = z/-5
Đặt x/-2= y/3 = z/-5 = k
=> x= -2k ; y= 3k ; z= -5k
=> xyz = 30. k^3 = 30000 => k^3 = 1000 => k = 10
=> x= -20 ; y = 30 ; z= -50
Vậy x= -20 ; y= 30 ; z= -50
\(15x=10y=6z\Leftrightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}=\frac{2x-z}{2.2-5}=\frac{5}{-1}=-5\)
\(\Rightarrow y=-5.3=-15\)