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a: \(\left\{{}\begin{matrix}\dfrac{2}{x}+\dfrac{3}{y}=5\\\dfrac{1}{x}-\dfrac{4}{y}=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{x}+\dfrac{3}{y}=5\\\dfrac{2}{x}-\dfrac{8}{y}=-6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{11}{y}=11\\\dfrac{1}{x}-\dfrac{4}{y}=-3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=1\\\dfrac{1}{x}=-3+\dfrac{4}{y}=-3+4=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=1\end{matrix}\right.\)
b: \(\left\{{}\begin{matrix}\dfrac{12}{x-3}-\dfrac{5}{y+2}=63\\\dfrac{8}{x-3}+\dfrac{15}{y+2}=-13\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{36}{x-3}-\dfrac{15}{y+2}=189\\\dfrac{8}{x-3}+\dfrac{15}{y+2}=-13\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{44}{x-3}=176\\\dfrac{8}{x-3}+\dfrac{15}{y+2}=-13\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-3=\dfrac{1}{4}\\\dfrac{15}{y+2}=-13-\dfrac{8}{x-3}=-13-32=-45\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{13}{4}\\y=-\dfrac{1}{3}-2=-\dfrac{7}{3}\end{matrix}\right.\)
a)\(\left\{{}\begin{matrix}2x-3y=1\\x+2y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2\cdot\left(3-2y\right)-3y=1\\x=3-2y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6-7y=1\\x=3-2y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{5}{7}\\x=3-2\cdot\dfrac{5}{7}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{5}{7}\\x=\dfrac{11}{7}\end{matrix}\right.\)b) Biểu diễn lại một biến theo một biến như pt trên rồi giải, ta có :
\(\left\{{}\begin{matrix}2x+4y=5\\4x-2y=2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=\dfrac{9}{10}\\y=\dfrac{4}{5}\end{matrix}\right.\)
c) Cách làm tương tự như pt a ta có :
\(\left\{{}\begin{matrix}\dfrac{2}{3}x+\dfrac{1}{2}y=\dfrac{2}{3}\\\dfrac{1}{3}x-\dfrac{3}{4}y=\dfrac{1}{2}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=\dfrac{9}{8}\\y=-\dfrac{1}{6}\end{matrix}\right.\)
d) Tương tự ta có :
\(\left\{{}\begin{matrix}0,3x-0,2y=0,5\\0,5x+0,4y=1,2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=2\\y=\dfrac{1}{2}\end{matrix}\right.\)
Lời giải:
\(\left\{\begin{matrix} x+\frac{1}{y}=2(1)\\ y+\frac{1}{z}=2(2)\\ z+\frac{1}{x}=2(3)\end{matrix}\right.\)
Lấy \((1)-(2); (2)-(3); (3)-(1)\) ta thu được:
\(\left\{\begin{matrix} x-y+\frac{z-y}{yz}=0\\ y-z+\frac{x-z}{xz}=0\\ z-x+\frac{y-x}{xy}=0\end{matrix}\right.\) \(\Leftrightarrow \left\{\begin{matrix} x-y=\frac{y-z}{yz}\\ y-z=\frac{z-x}{xz}\\ z-x=\frac{x-y}{xy}\end{matrix}\right.\)
\(\Rightarrow (x-y)(y-z)(z-x)=\frac{(x-y)(y-z)(z-x)}{(xyz)^2}\)
\(\Leftrightarrow (x-y)(y-z)(z-x)(1-\frac{1}{xyz})(1+\frac{1}{xyz})=0\)
TH1: \(x-y=0\Leftrightarrow x=y\Rightarrow x+\frac{1}{x}=2\)
\(\Rightarrow x^2-2x+1=0\Leftrightarrow (x-1)^2=0\Leftrightarrow x=1\rightarrow y=1\)
Thay vào PT\((2)\Rightarrow 1+\frac{1}{z}=2\rightarrow z=1\)
Ta thu được \((x,y,z)=(1,1,1)\)
TH2: \(y-z=0; z-x=0\) hoàn toàn giống TH1 ta cũng có \((x,y,z)=(1,1,1)\)
TH3: \(1-\frac{1}{xyz}=1\Rightarrow xyz=1\)
Thay vào PT(1) và (2)
\(\left\{\begin{matrix} x+\frac{1}{y}=2\\ y+xy=2\end{matrix}\right.\Rightarrow \left\{\begin{matrix} xy+1=2y\\ xy=2-y\end{matrix}\right.\)
\(\Rightarrow 2-y+1=2y\Leftrightarrow y=1\Rightarrow x=z=1\)
TH4: \(1+\frac{1}{xyz}=0\Leftrightarrow xyz=-1\)
Thay vào PT (1) và (2):
\(\left\{\begin{matrix} x+\frac{1}{y}=2\\ y-xy=2\end{matrix}\right.\Rightarrow \left\{\begin{matrix} xy+1=2y\\ xy=y-2\end{matrix}\right.\)
\(\Rightarrow y-2+1=2y\Leftrightarrow y=-1\)
\(\Rightarrow x+\frac{1}{-1}=2\Rightarrow x=3; -1+\frac{1}{z}=2\Rightarrow z=\frac{1}{3}\)
Thử vào PT(3) thấy không đúng (loại)
Vậy \((x,y,z)=(1,1,1)\)
1. \(\left\{{}\begin{matrix}x+y+\dfrac{1}{x}+\dfrac{1}{y}=5\\x^2+y^2+\dfrac{1}{x^2}+\dfrac{1}{y^2}=9\end{matrix}\right.\) ĐKXĐ : \(\left\{{}\begin{matrix}x>0\\y>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2y+xy^2+x+y=5xy\\x^4y^2+x^2y^4+x^2+y^2=9x^2y^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x^4y^2+x^2y^4+x^2+y^2=25x^2y^2\\x^4y^2+x^2y^4+x^2+y^2=9x^2y^2\end{matrix}\right.\)\(\Leftrightarrow0=16x^2y^2\)
\(\Rightarrow\) phương trình vô nghiệm
\(\left\{{}\begin{matrix}x+y+z=1\\\left(1+\dfrac{1}{x}\right)\left(1+\dfrac{1}{y}\right)\left(1+\dfrac{1}{z}\right)=64\end{matrix}\right.\)
Ta có:
\(1=x+y+z\ge3\sqrt[3]{xyz}\)
\(\Leftrightarrow xyz\le\dfrac{1}{27}\)
Ta có: \(\left(1+\dfrac{1}{x}\right)\left(1+\dfrac{1}{y}\right)\left(1+\dfrac{1}{z}\right)=1+\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{zx}\right)+\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)+\dfrac{1}{xyz}\)
\(\ge1+\dfrac{3}{\sqrt[3]{x^2y^2z^2}}+\dfrac{3}{\sqrt[3]{xyz}}+\dfrac{1}{xyz}\)
\(=1+\dfrac{3}{\sqrt[3]{\dfrac{1}{27^2}}}+\dfrac{3}{\sqrt[3]{\dfrac{1}{27}}}+\dfrac{1}{\dfrac{1}{27}}=64\)
Dấu = xảy ra khi \(x=y=z=\dfrac{1}{3}\)
Phần cuối là:
\(\ge1+\dfrac{3}{\sqrt[3]{\dfrac{1}{27}}}+\dfrac{3}{\sqrt[3]{\dfrac{1}{27}}}+\dfrac{1}{\dfrac{1}{27}}=64\), không phải là dấu ''=''.
a) \(\left\{{}\begin{matrix}5x+3y=-7\\2x-4y=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5x+3y=-7\\x-2y=3\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}5x+3y=-7\\x=3+2y\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5.\left(3+2y\right)+3y=-7\\x=3+2y\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}13y=-22\\x=3+2y\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=3+2.\dfrac{-22}{13}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=\dfrac{-5}{13}\end{matrix}\right.\)
Vậy hệ phương trình có nghiệm là: \(\left\{{}\begin{matrix}y=\dfrac{-22}{13}\\x=\dfrac{-5}{13}\end{matrix}\right.\).
b)\(\left\{{}\begin{matrix}7x+14y=17\\2x+4y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}14x+28y=34\\14x+28y=35\end{matrix}\right.\) (vô nghiệm)
Vậy hệ phương trình vô nghiệm.
lời giải
a) \(\left\{{}\begin{matrix}-2x+\dfrac{3}{5}>\dfrac{2x-7}{3}\left(1\right)\\x-\dfrac{1}{2}< \dfrac{5\left(3x-1\right)}{2}\left(2\right)\end{matrix}\right.\)
(1)\(\Leftrightarrow\)
\(\dfrac{3}{5}+\dfrac{7}{3}>\left(\dfrac{2}{3}+2\right)x\)
\(\dfrac{44}{15}>\dfrac{8}{3}x\) \(\Rightarrow x< \dfrac{44.3}{15.8}=\dfrac{11}{5.2}=\dfrac{11}{10}\)
Nghiêm BPT(1) là \(x< \dfrac{11}{10}\)
(2) \(\Leftrightarrow2x-1< 15x-5\Rightarrow13x>4\Rightarrow x>\dfrac{4}{13}\)
Ta có: \(\dfrac{4}{13}< \dfrac{11}{10}\) => Nghiệm hệ (a) là \(\dfrac{4}{13}< x< \dfrac{11}{10}\)
Cái đề có phải là
\(\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=5\\\dfrac{2}{xy}-\dfrac{1}{z^2}=25\end{matrix}\right.????\)