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Câu 1:
c: 2x=3y
nên x/3=y/2
=>x/9=y/6
5y=3z
nên y/3=z/5
=>y/6=z/10
=>x/9=y/6=z/10
Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:
\(\dfrac{x}{9}=\dfrac{y}{6}=\dfrac{z}{10}=\dfrac{3x+3y-7z}{3\cdot9+3\cdot6-7\cdot10}=\dfrac{35}{-25}=-\dfrac{7}{5}\)
Do đó: x=-63/5; y=-42/5; z=-14
Bài 2:
Gọi ba số lần lượt là a,b,c
Theo đề, ta có: 4/3a=b=3/4c
\(\Leftrightarrow\dfrac{a}{\dfrac{3}{4}}=\dfrac{b}{1}=\dfrac{c}{\dfrac{4}{3}}\)
\(\Leftrightarrow\dfrac{a}{9}=\dfrac{b}{12}=\dfrac{c}{16}\)
Đặt \(\dfrac{a}{9}=\dfrac{b}{12}=\dfrac{c}{16}=k\)
=>a=9k; b=12k; c=16k
Theo đề, ta có: \(a^2+b^2+c^2=481\)
\(\Leftrightarrow81k^2+144k^2+256k^2=481\)
=>k2=1
Trường hợp 1: k=1
=>a=9; b=12; c=16
Trường hợp 2: k=-1
=>a=-9; b=-12; c=-16
\(A=\left(\dfrac{-3}{7}.x^3.y^2\right).\left(\dfrac{-7}{9}.y.z^2\right).\left(6.x.y\right)\)
\(A=\left(\dfrac{-3}{7}x^3y^2\right).\left(\dfrac{-7}{9}yz^2\right).6xy\)
\(A=\left(\dfrac{-3}{7}.\dfrac{-7}{9}.6\right).\left(x^3.x\right)\left(y^2.y.y\right).z^2\)
\(A=2x^4y^4z^2\)
\(B=-4.x.y^3\left(-x^2.y\right)^3.\left(-2.x.y.z^3\right)^2\)
\(B=\left[\left(-4\right).\left(-2\right)\right].\left(x.x^6.x^2\right)\left(y^3.y^3.y^2\right)\left(z^6\right)\)
\(B=8x^7y^{y^8}z^6\)
a: Đặt a/b=c/d=k
=>a=bk; c=dk
\(\left(\dfrac{a+b}{c+d}\right)^2=\left(\dfrac{bk+b}{dk+d}\right)^2=\dfrac{b^2}{d^2}\)
\(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{b^2k^2+b^2}{d^2k^2+d^2}=\dfrac{b^2}{d^2}\)
Do đó: \(\left(\dfrac{a+b}{c+d}\right)^2=\dfrac{a^2+b^2}{c^2+d^2}\)
b: \(\dfrac{ab}{cd}=\dfrac{bk\cdot b}{dk\cdot d}=\dfrac{b^2}{d^2}\)
\(\left(\dfrac{a-b}{c-d}\right)^2=\left(\dfrac{bk-b}{dk-d}\right)^2=\dfrac{b^2}{d^2}\)
Do đó: \(\dfrac{ab}{cd}=\left(\dfrac{a-b}{c-d}\right)^2\)
Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\Rightarrow\left\{{}\begin{matrix}a=b.k\\c=d.k\end{matrix}\right.\)
Ta có:
\(\dfrac{4.a-5.b}{4.a+5.b}=\dfrac{4.a+5.b-10.b}{4.a+5.b}=1-\dfrac{10.b}{4.a+5.b}=1-\dfrac{10.b}{4.b.k+5b}=1-\dfrac{10}{4.k+5}\) (1)
\(\dfrac{4.c-5.d}{4.c+5.d}=\dfrac{4.c+5.d-10.d}{4.c+5.d}=1-\dfrac{10.d}{4.c+5.d}=1-\dfrac{10.d}{4.d.k+5.d}=1-\dfrac{10}{4.k+5}\) (2)
Từ (1) và (2) suy ra \(\dfrac{4.a-5.b}{4.a+5.b}=\dfrac{4.c-5.d}{4.c+5.d}\left(đpcm\right)\)
Lời giải:
Đặt \(\frac{a}{b}=\frac{c}{d}=t\Rightarrow a=bt; c=dt\)
Khi đó ta có:
\(\frac{4a-5b}{4a+5b}=\frac{4bt-5b}{4bt+5b}=\frac{b(4t-5)}{b(4t+5)}=\frac{4t-5}{4t+5}\)
\(\frac{4c-5d}{4c+5d}=\frac{4dt-5d}{4dt+5d}=\frac{d(4t-5)}{d(4t+5)}=\frac{4t-5}{4t+5}\)
Do đó: \(\frac{4a-5b}{4a+5b}=\frac{4c-5d}{4c+5d}\) (đpcm)
Ta có : \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=\dfrac{xyz}{2\cdot3\cdot5}=\dfrac{800}{30}=\dfrac{80}{3}\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{80}{3}\Rightarrow x=\dfrac{160}{3}\) ∼ \(53\)
\(\Rightarrow\dfrac{y}{3}=\dfrac{80}{3}\Rightarrow y=80\)
\(\Rightarrow\dfrac{z}{5}=\dfrac{80}{3}\Rightarrow z=\dfrac{400}{3}\) ∼ 133
Mk xl, bài lúc nãy mk lm là sai, đây ms là bài đúng:
Theo bài ra ta có : \(\left\{{}\begin{matrix}\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}\\xyz=800\left(1\right)\end{matrix}\right.\)
Đặt \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{5}=k\)
\(\Rightarrow\left\{{}\begin{matrix}x=2k\\y=3k\\z=5k\end{matrix}\right.\) (2)
Thay (2) vào ( 1) ta có : \(2k\cdot3k\cdot5k=800\)
\(.....................................................................\)
Rồi cứ tìm ra \(k\) rồi thay \(k\) vào mà tính \(x,y,z\) bth thôi bạn ạ
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{a+b-c}{3c}=\dfrac{b+c-a}{3a}=\dfrac{c+a-b}{3b}=\dfrac{a+b-c+b+c-a+c+a-b}{3a+3b+3c}=\dfrac{a+b+c+\left(a-a\right)+\left(b-b\right)+\left(c-c\right)}{3a+3b+3c}=\dfrac{a+b+c}{3\left(a+b+c\right)}=\dfrac{1}{3}\)
Khi đó:
\(\left\{{}\begin{matrix}\dfrac{a+b-c}{3c}=\dfrac{1}{3}\\\dfrac{b+c-a}{3a}=\dfrac{1}{3}\\\dfrac{c+a-b}{3b}=\dfrac{1}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3a+3b-3c=3c\\3b+3c-3a=3a\\3c+3a-3b=3b\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3a+3b=6c\\3b+3c=6a\\3c+3a=6b\end{matrix}\right.\)Thay vào \(P\)
\(P=\left(1+\dfrac{b}{a}\right)\left(1+\dfrac{a}{c}\right)\left(1+\dfrac{c}{b}\right)=\left(\dfrac{a+b}{a}\right)\left(\dfrac{c+a}{c}\right)\left(\dfrac{b+c}{b}\right)\)
\(27P=3\left(\dfrac{a+b}{a}\right).3\left(\dfrac{c+a}{c}\right).3\left(\dfrac{b+c}{b}\right)\)
\(27P=\left(\dfrac{3a+3b}{a}\right)\left(\dfrac{3c+3a}{c}\right)\left(\dfrac{3b+3c}{b}\right)\)
\(27P=\)\(\dfrac{6c}{a}.\dfrac{6b}{c}.\dfrac{6a}{b}=\dfrac{216abc}{abc}=216\Leftrightarrow P=\dfrac{216}{27}=8\)
a)x=1;2;-2(bạn nên tự giải)
b)=>\(\dfrac{1\cdot2\cdot3\cdot4\cdot...\cdot30\cdot31}{4\cdot6\cdot8\cdot10\cdot...\cdot62\cdot64}\)=2x
=>\(\dfrac{2\cdot3\cdot4\cdot5\cdot...\cdot30\cdot31}{60\left(2\cdot3\cdot4\cdot5\cdot...\cdot30\cdot31\right)\cdot64}=2x\)
=>\(\dfrac{1}{60\cdot64}=2x\)=> 1/3840 =2x
=>x = 1/7680
c)=>4x - 2x = 6x - 3x
=>2x (2x-1)= 3x(2x-1)
=> 2x = 3x
=>x = 0
Sửa đề: Cho \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\) . CMR: \(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)
Giải:
\(\dfrac{b.z-x.y}{a}=\dfrac{c.x-a.z}{b}=\dfrac{a.y-b.x}{c}\)
\(\Rightarrow\dfrac{a\left(bz-cy\right)}{a^2}=\dfrac{b\left(cx-az\right)}{b^2}=\dfrac{c\left(ay-bz\right)}{c^2}\)
\(\Rightarrow\dfrac{abz-acy}{a^2}=\dfrac{bcx-abz}{b^2}=\dfrac{acy-bcx}{c^2}\)
\(\Rightarrow\dfrac{abz-acy+bcx-abz+acy-bcx}{a^2+b^2+c^2}\)
\(\Rightarrow\dfrac{0}{a^2+b^2+c^2}\)
\(=0\)
\(\dfrac{bz-cy}{a}=0\)
\(\Rightarrow bz-cy=0\)
\(\Rightarrow\dfrac{z}{c}=\dfrac{y}{b}\left(1\right)\)
\(\dfrac{cx-az}{b}=0\)
\(\Rightarrow cx-az=0\)
\(\Rightarrow cx=az\)
\(\Rightarrow\dfrac{x}{a}=\dfrac{z}{c}\left(2\right)\)
Từ (1) và (2) suy ra:
\(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)