Giải hệ pt sau \(\left\{{}\begin{matrix}x^2-xy+y^2=3\\z^2+yz+1=0\end{matrix}\right.\)

Lên mặt làm gì, làm gì lên được -_-

\(\left\{{}\begin{matrix}\sqrt{5x^2+2xy+2y^2}+\sqrt{2x^2+2xy+5y^2}=3\left(x+y\right)\\\sqrt{2x+y+1}+2\sqrt[3]{7x+12y+8}=2xy+y+5\end{matrix}\right.\)

Xét \(pt\left(1\right)\) dễ dàng suy ra \(x+y\ge0\)

\(VT=\sqrt{\left(x-y\right)^2+\left(2x+y\right)^2}+\sqrt{\left(x-y\right)^2+\left(2y+x\right)^2}\)

\(\ge\left|2x+y\right|+\left|2y+x\right|\ge3\left(x+y\right)\)

Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x=y\\x,y\ge0\end{matrix}\right.\)

Thay vào \(pt\left(2\right)\) ta được:

\(\sqrt{3x+1}+2\sqrt[3]{19x+8}=2x^2+x+5\)

\(\Leftrightarrow\left[\sqrt{3x+1}-\left(x+1\right)\right]+2\left[\sqrt[3]{19x+8}-\left(x+2\right)\right]=2x^2-2x\)

\(\Leftrightarrow\left(x-x^2\right)\left[\dfrac{1}{\sqrt{3x+1}+x+1}+2\cdot\dfrac{x+7}{\sqrt[3]{\left(19x+8\right)^2}+\left(x+2\right)\sqrt[3]{19x+8}+\left(x+2\right)^2}+2\right]=0\)

Do \(x;y\ge0\) nên pt trong ngoặc luôn dương

\(\Rightarrow x-x^2=0\Rightarrow x\left(1-x\right)=0\Rightarrow\)\(\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)

Mà \(x=y\)\(\Rightarrow\left[{}\begin{matrix}x=y=0\\x=y=1\end{matrix}\right.\) là nghiệm của hpt

thanks b đã chỉ giúp mình.tại đánh máy nên mình ko để ý^^

pt(1): 5x²+2xy+2y²>=(2x+y)² nên \(\sqrt{5x^{2^{ }}+2xy+2y^2}\ge\:\)trị tuyệt đối 2x+y.

cmtt>\(\sqrt{2x^2+2xy+5y^2}\ge\)trị tuyệt đối x+ 2y.

>mà tt đối 2x+y cộng ttđ x+2y>= 3(x+y).

>(1)>=3(x+y).

đâu = xảy ra khi và chỉ khi x=y.

thay x=y >=0 vào (2):

\(\sqrt{3x+1}+2\sqrt[3]{19x+8}\) = 2x²+x+5.

<=>\(\left(\sqrt{3x+1}-\left(x+1\right)\right)\)+\(\left(2\sqrt[3]{19x+8}-\left(x+2\right)\right)\)= 2x²- 2x.

nhân liên hợp ta đc:

(x²-x)*(\(\dfrac{1}{\sqrt{3x+1}+x+1}+2\dfrac{x+7}{\sqrt[3]{19x+18}+\left(x+2\right)\left(\sqrt[3]{19x+18}\right)+\left(x+2\right)^2}=0\)

dễ thấy phần *>0 với mọi x,ytheo đk của (1)

>(x² -x)=0

>x=0 hoặc x=1

>(x,y)=(0,0); (1,1).

vậy....

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+y\right)^2-2xy=10\\xy\left(x+y\right)+5\left(x+y\right)=32\end{matrix}\right.\)

Đặt \(\left\{{}\begin{matrix}x+y=u\\xy=v\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}u^2-2v=10\\uv+5u=32\end{matrix}\right.\)

\(\Rightarrow u\left(\dfrac{u^2-10}{2}\right)+5u=32\)

\(\Leftrightarrow u^3=64\Rightarrow u=4\Rightarrow v=3\)

\(\Rightarrow\left(x;y\right)=\left(1;3\right);\left(3;1\right)\)

a) \(\left\{{}\begin{matrix}2x+3y=5\\4x-5y=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}4x+6y=10\\4x-5y=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x+3y=5\\11y=9\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x+3\cdot\dfrac{9}{11}=5\\y=\dfrac{9}{11}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x+\dfrac{27}{11}=5\\y=\dfrac{9}{11}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x=\dfrac{28}{11}\\y=\dfrac{9}{11}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{14}{11}\\y=\dfrac{9}{11}\end{matrix}\right.\)

Vậy: \(x=\dfrac{14}{11};y=\dfrac{9}{11}\)

\(a,\Leftrightarrow\left\{{}\begin{matrix}5x+15y=-10\\5x-4y=11\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}19y=-21\\5x-4y=11\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{21}{19}\\5x-4\left(-\dfrac{21}{19}\right)=11\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{25}{19}\\y=-\dfrac{21}{19}\end{matrix}\right.\)

\(c,\Leftrightarrow\left\{{}\begin{matrix}3x+5y=1\\10x-5y=-40\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+5y=1\\13x=-39\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=2\end{matrix}\right.\\ d,\Leftrightarrow\left\{{}\begin{matrix}5x-10y=-30\\5x-3y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5x-3y=5\\-7y=-35\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=4\\y=5\end{matrix}\right.\\ e,\Leftrightarrow\left\{{}\begin{matrix}2\left(x+y\right)+3\left(x-y\right)=4\\2\left(x+y\right)+4\left(x-y\right)=10\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y=6\\2\left(x+y\right)+3\cdot6=4\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x-y=6\\x+y=-7\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{1}{2}\\y=-\dfrac{13}{2}\end{matrix}\right.\)

a: \(\left\{{}\begin{matrix}x+4y=-11\\5x-4y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6x=-10\\x+4y=-11\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{-5}{3}\\y=\dfrac{-11-x}{4}=\dfrac{-11+\dfrac{5}{3}}{4}=-\dfrac{7}{3}\end{matrix}\right.\)

b: \(\left\{{}\begin{matrix}2x-y=7\\3x+5y=-22\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6x-3y=21\\6x+15y=-66\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-18y=78\\2x-y=7\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-13}{3}\\x=\dfrac{y+7}{2}=\dfrac{4}{3}\end{matrix}\right.\)

<=> xy+5x+3y+15=xy+8x+y+8                 <=> 3x-2y=7           <=>  9x-6y=21 <=> x=3            <=> x=3

      10xy+14x-15y-21=10xy+10x-12y-12            4x-3y=9                  8x-6y=18       8.3-6y=18           y=1

moi hok lop 6 thoi

mới hok 5 thôi