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a, \(\left\{{}\begin{matrix}x+y=4\\\left(x^2+y^2\right)\left(x^3+y^3\right)=280\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=4\\\left(x^2+y^2\right)\left(x^2+y^2-xy\right)=70\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=4\\\left(16-2xy\right)\left(16-3xy\right)=70\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=4\\3x^2y^2-40xy+93=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=4\\\left[{}\begin{matrix}xy=\dfrac{31}{3}\\xy=3\end{matrix}\right.\end{matrix}\right.\)
TH1: \(\left\{{}\begin{matrix}x+y=4\\xy=3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=1\\y=3\end{matrix}\right.\\\left\{{}\begin{matrix}x=3\\y=1\end{matrix}\right.\end{matrix}\right.\)
TH2: \(\left\{{}\begin{matrix}x+y=4\\xy=\dfrac{31}{3}\end{matrix}\right.\)
Phương trình này vô nghiệm
Vậy hệ đã cho có nghiệm \(\left(x;y\right)\in\left\{\left(1;3\right);\left(3;1\right)\right\}\)
b, ĐK: \(xy>0\)
\(\left\{{}\begin{matrix}\sqrt{\dfrac{2x}{y}}+\sqrt{\dfrac{2y}{x}}=3\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2x}{y}+\dfrac{2y}{x}+4=9\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2\left(x^2+y^2\right)=5xy\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(2x-y\right)\left(x-2y\right)=0\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}2x=y\\x=2y\end{matrix}\right.\\x-y+xy=3\end{matrix}\right.\)
TH1: \(\left\{{}\begin{matrix}y=2x\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=2x\\2x^2-x-3=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=2x\\\left(x+1\right)\left(2x-3\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}y=-2\\x=-1\end{matrix}\right.\\\left\{{}\begin{matrix}y=3\\x=\dfrac{3}{2}\end{matrix}\right.\end{matrix}\right.\)
TH2: \(\left\{{}\begin{matrix}x=2y\\x-y+xy=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=2y\\2y^2+y-3=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=3\\y=\dfrac{3}{2}\end{matrix}\right.\end{matrix}\right.\)
Vậy ...
a.
\(\left\{{}\begin{matrix}x^3-y^3=16x-4y\\-4=5x^2-y^2\end{matrix}\right.\)
Nhân vế:
\(-4\left(x^3-y^3\right)=\left(16x-4y\right)\left(5x^2-y^2\right)\)
\(\Leftrightarrow21x^3-5x^2y-4xy^2=0\)
\(\Leftrightarrow x\left(7x-4y\right)\left(3x+y\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{4y}{7}\\y=-3x\end{matrix}\right.\)
Thế vào \(y^2=5x^2+4...\)
b. Đề bài không hợp lý ở \(4x^2\)
c.
\(\Leftrightarrow\left\{{}\begin{matrix}x^3-y^3=9\\3x^2+6y^2=3x-12y\end{matrix}\right.\)
Trừ vế:
\(x^3-y^3-3x^2-6y^2=9-3x+12y\)
\(\Leftrightarrow x^3-3x^2+3x-1=y^3+6y^2+12y+8\)
\(\Leftrightarrow\left(x-1\right)^3=\left(y+2\right)^3\)
\(\Leftrightarrow x-1=y+2\)
\(\Leftrightarrow y=x-3\)
Thế vào \(x^2=2y^2=x-4y\) ...
a.
Với \(y=0\) không phải nghiệm
Với \(y\ne0\Rightarrow\left\{{}\begin{matrix}3x+2=\dfrac{5}{y}\\2x\left(x+y\right)+y=\dfrac{5}{y}\end{matrix}\right.\)
\(\Rightarrow3x+2=2x\left(x+y\right)+y\)
\(\Leftrightarrow2x^2+\left(2y-3\right)x+y-2=0\)
\(\Delta=\left(2y-3\right)^2-8\left(y-2\right)=\left(2y-5\right)^2\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{-2y+3+2y-5}{4}=-\dfrac{1}{2}\\x=\dfrac{-2y+3-2y+5}{4}=-y+2\end{matrix}\right.\)
Thế vào pt đầu ...
Câu b chắc chắn đề sai
a, \(\left\{{}\begin{matrix}\left(x-y\right)\left(x^2+y^2\right)=13\\\left(x+y\right)\left(x^2-y^2\right)=25\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2\left(x-y\right)\left(x^2+y^2\right)=26\\\left(x-y\right)\left(x+y\right)^2=25\end{matrix}\right.\)
Trừ vế theo vế \(pt\left(1\right)\) cho \(pt\left(2\right)\) ta được:
\(\Leftrightarrow\left(x-y\right)\left(x^2+y^2-2xy\right)=1\)
\(\Leftrightarrow\left(x-y\right)^3=1\)
\(\Leftrightarrow x-y=1\)
Khi đó hệ trở thành:
\(\left\{{}\begin{matrix}x^2+y^2=13\\\left(x+y\right)^2=25\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=13\\13+2xy=25\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2=13\\2xy=12\end{matrix}\right.\)
Cộng vế theo vế 2 phương trình:
\(\left(x+y\right)^2=25\)
\(\Leftrightarrow x+y=\pm5\)
TH1: \(x+y=5\)
Ta có hệ: \(\left\{{}\begin{matrix}x-y=1\\x+y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=2\end{matrix}\right.\)
TH2: \(x+y=-5\)
Ta có hệ: \(\left\{{}\begin{matrix}x-y=1\\x+y=-5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-2\\y=-3\end{matrix}\right.\)
b, \(\left\{{}\begin{matrix}2x^2+x-\dfrac{1}{y}=2\\y-y^2x-2y^2=-2\end{matrix}\right.\)
ĐK: \(y\ne0\)
\(\left\{{}\begin{matrix}2x^2+x-\dfrac{1}{y}=2\\y-y^2x-2y^2=-2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x^2+x-\dfrac{1}{y}=2\\\dfrac{1}{y}-x-2=-\dfrac{2}{y^2}\end{matrix}\right.\)
Đặt \(\dfrac{1}{y}=t\), hệ trở thành:
\(\Leftrightarrow\left\{{}\begin{matrix}2x^2+x-t=2\\2t^2+t-x=2\end{matrix}\right.\)
\(\Rightarrow\left(x-t\right)\left(x+t+1\right)=0\)
\(\Leftrightarrow...\)
Đây chắc chắn là 1 hệ pt không giải được
Lần lượt lấy (trên + dưới) và lấy (dưới - trên) được 1 hệ mới, sau đó chia vế cho vế và đặt \(\dfrac{x}{y}=t\) sẽ đưa về 1 pt không thể phân tích thành nhân tử, đồng nghĩa không thể giải hệ đã cho
bài ni đúng đề thầy ạ !
nghiệm của hệ pt là :\(\left(x,y\right)=\left\{\dfrac{1+\sqrt[5]{3}}{2},\dfrac{\sqrt[5]{3}-1}{2}\right\}\)
a.
\(\Leftrightarrow\left\{{}\begin{matrix}x^3-448y^3=-3x+6y\\96=385x^2-16y^2\end{matrix}\right.\)
\(\Rightarrow96\left(x^3-448y^3\right)=\left(-3x+6y\right)\left(385x^2-16y^2\right)\)
\(\Leftrightarrow\left(x-4y\right)\left(417x^2+898xy+3576y^2\right)=0\)
\(\Leftrightarrow x-4y=0\)
\(\Leftrightarrow x=4y\)
Thế vào \(385x^2-16y^2=96\)
\(\Rightarrow...\)
b.
ĐKXĐ: \(x+y\ne0\)
\(\left\{{}\begin{matrix}\left(3x^3-y^3\right)\left(x+y\right)=1\\1=x^2+y^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(3x^3-y^3\right)\left(x+y\right)=1\\1=\left(x^2+y^2\right)^2\end{matrix}\right.\)
\(\Rightarrow\left(3x^3-y^3\right)\left(x+y\right)=\left(x^2+y^2\right)^2\)
\(\Leftrightarrow\left(x-y\right)\left(x+2y\right)\left(2x^2+xy+y^2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=y\\x=-2y\end{matrix}\right.\)
Thế vào \(x^2+y^2=1\)...
\(x-\dfrac{1}{x}=y-\dfrac{1}{y}\Leftrightarrow x-y+\dfrac{1}{y}-\dfrac{1}{x}=0\)
\(\Leftrightarrow x-y+\dfrac{x-y}{xy}=0\Leftrightarrow\left(x-y\right)\left(1+\dfrac{1}{xy}\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}y=x\\y=-\dfrac{1}{x}\end{matrix}\right.\)
Thế vào pt dưới:
\(\left[{}\begin{matrix}x^3+1=2x\\x^3+1=-\dfrac{2}{x}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\left(x-1\right)\left(x^2+x-1\right)=0\\x^4+x+2=0\Leftrightarrow\left(x^2-\dfrac{1}{2}\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{2}=0\left(vn\right)\end{matrix}\right.\)
ý a ở đây bn https://hoc247.net/hoi-dap/toan-10/giai-he-pt-3x-x-2-2-y-2-va-3y-y-2-2-x-2-faq371128.html
b.
Với \(xy=0\) không là nghiệm
Với \(xy\ne0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\left(y^2+1\right)=y\left(5-x^2\right)\\y^2+1=y\left(5-2x\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{y^2+1}{y}=\dfrac{5-x^2}{x}\\\dfrac{y^2+1}{y}=5-2x\end{matrix}\right.\)
\(\Rightarrow\dfrac{5-x^2}{x}=5-2x\)
\(\Leftrightarrow5-x^2=5x-2x^2\)
\(\Leftrightarrow...\)
\(\left\{{}\begin{matrix}x^3y^2+x^2y^3+x^3y+2x^2y^2+xy^3-30=0\\x^2y+xy^2+xy+x+y-11=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2y^2\left(x+y\right)+xy\left(x+y\right)^2-30=0\\xy\left(x+y\right)+xy+x+y-11=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}xy\left(x+y\right)\left[xy+x+y\right]-30=0\\xy\left(x+y\right)+xy+x+y-11=0\end{matrix}\right.\)
Đặt \(\left\{{}\begin{matrix}xy\left(x+y\right)=u\\xy+x+y=v\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}uv-30=0\\u+v-11=0\end{matrix}\right.\) \(\Rightarrow\left(u;v\right)=\left(6;5\right);\left(5;6\right)\)
TH1: \(\left\{{}\begin{matrix}xy\left(x+y\right)=6\\xy+x+y=5\end{matrix}\right.\)
Theo Viet đảo \(\Rightarrow\left\{{}\begin{matrix}x+y=3\\xy=2\end{matrix}\right.\) \(\Rightarrow\left(x;y\right)=\left(1;2\right);\left(2;1\right)\)hoặc \(\left\{{}\begin{matrix}x+y=2\\xy=3\end{matrix}\right.\)(vô nghiệm)
TH2: \(\left\{{}\begin{matrix}xy\left(x+y\right)=5\\xy+x+y=6\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x+y=5\\xy=1\end{matrix}\right.\) \(\Rightarrow...\) hoặc \(\left\{{}\begin{matrix}x+y=1\\xy=5\end{matrix}\right.\) (vô nghiệm)
2 câu dưới hình như em hỏi rồi?
ĐK: \(x,y\ne0\)
\(\left\{{}\begin{matrix}x-\dfrac{1}{x^3}=y-\dfrac{1}{y^3}\\\left(x-4y\right)\left(2x-y+4\right)=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-y-\left(\dfrac{1}{x^3}-\dfrac{1}{y^3}\right)=0\\\left(x-4y\right)\left(2x-y+4\right)=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-y+\dfrac{\left(x-y\right)\left(x^2+y^2+xy\right)}{x^3y^3}=0\\\left(x-4y\right)\left(2x-y+4\right)=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{\left(x-y\right)\left(x^3y^3+x^2+y^2+xy\right)}{x^3y^3}=0\\\left(x-4y\right)\left(2x-y+4\right)=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\\left(x-3x\right)\left(2x-x+4\right)=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\-2x^2-8x=-36\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\x^2+4x-18=0\end{matrix}\right.\)
\(\Leftrightarrow x=y=-2\pm\sqrt{22}\left(tm\right)\)