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b)\(\sqrt{5x^2+2xy+2y^2}+\sqrt{2x^2+2xy+5y^2}=3\left(x+y\right)\)
\(\Rightarrow\left(\sqrt{5x^2+2xy+2y^2}+\sqrt{2x^2+2xy+5y^2}\right)^2=\left(3\left(x+y\right)\right)^2\)
\(\Leftrightarrow\sqrt{\left(5x^2+2xy+2y^2\right)\left(2x^2+2xy+5y^2\right)}=x^2+7xy+y^2\)
\(\Rightarrow\left(5x^2+2xy+2y^2\right)\left(2x^2+2xy+5y^2\right)=\left(x^2+7xy+y^2\right)^2\)
\(\Leftrightarrow9\left(x-y\right)^2\left(x+y\right)^2=0\)\(\Leftrightarrow\left[{}\begin{matrix}x=y\\x=-y\end{matrix}\right.\)
\(\rightarrow\left(x;y\right)\in\left\{\left(0;0\right),\left(1;1\right)\right\}\)
a: Đặt |x-6|=a, |y+1|=b
Theo đề, ta có hệ phương trình:
\(\left\{{}\begin{matrix}2a+3b=5\\5a-4b=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=1\end{matrix}\right.\)
=>|x-6|=1 và |y+1|=1
\(\Leftrightarrow\left\{{}\begin{matrix}x\in\left\{7;5\right\}\\y\in\left\{0;-2\right\}\end{matrix}\right.\)
b: Đặt |x+y|=a, |x-y|=b
Theo đề, ta có: \(\left\{{}\begin{matrix}2a-b=19\\3a+2b=17\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{55}{7}\\b=-\dfrac{23}{7}\left(loại\right)\end{matrix}\right.\)
=>HPTVN
c: Đặt |x+y|=a, |x-y|=b
Theo đề ta có: \(\left\{{}\begin{matrix}4a+3b=8\\3a-5b=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=2\\b=0\end{matrix}\right.\)
=>|x+y|=2 và x=y
=>|2x|=2 và x=y
=>x=y=1 hoặc x=y=-1
a.
\(\Leftrightarrow\left\{{}\begin{matrix}x\left(x^2+y^2\right)+\left(x^2+y^2-4\right)\left(y+2\right)=0\\x^2+y^2+\left(x+y-2\right)\left(y+2\right)=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(x^2+y^2-4\right)\left(y+2\right)=-x\left(x^2+y^2\right)\\-\left(x^2+y^2\right)=\left(x+y-2\right)\left(y+2\right)\end{matrix}\right.\)
\(\Rightarrow\left(x^2+y^2-4\right)\left(y+2\right)=x\left(x+y-2\right)\left(y+2\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}y+2=0\left(\text{không thỏa mãn}\right)\\x^2+y^2-4=x\left(x+y-2\right)\end{matrix}\right.\)
\(\Rightarrow x^2+y^2-4=x^2+x\left(y-2\right)\)
\(\Leftrightarrow\left(y+2\right)\left(y-2\right)=x\left(y-2\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}y=2\\x=y+2\end{matrix}\right.\)
Thế vào pt dưới:
\(\Rightarrow\left[{}\begin{matrix}x^2+8+2x+2x-4=0\\\left(y+2\right)^2+2y^2+y\left(y+2\right)+2\left(y+2\right)-4=0\end{matrix}\right.\)
\(\Leftrightarrow...\)
Câu b chắc chắn đề sai, nhìn 2 vế pt đầu đều có \(x^2\) thì chúng sẽ rút gọn, không ai cho đề như thế hết
b, Ta có : \(\left\{{}\begin{matrix}x^2-xy+3y^2+2x-5y-4=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x^2-xy+3y^2+2x-5y=4\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x^2-xy+3y^2+2x-5y=x+2y\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x^2-xy+3y^2+2x-5y-x-2y=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x^2-xy+3y^2+x-7y=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x^2+2xy+3y^2+1,5xy-4,5xy+x-7y=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x\left(x+2y\right)+1,5y\left(x+2y\right)-4,5xy+x-7y=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}4x+6y-4,5xy+x-7y=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}5x-y-4,5xy=0\\x+2y=4\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}5\left(4-2y\right)-y-4,5y\left(4-2y\right)=0\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}20-10y-y-18y+9y^2=0\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}20-29y+9y^2=0\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}9y^2-9y-20y+20=0\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}\left(9y-20\right)\left(y-1\right)=0\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}\left[{}\begin{matrix}y=1\\y=\frac{20}{9}\end{matrix}\right.\\x=4-2y\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}\left[{}\begin{matrix}y=1\\y=\frac{20}{9}\end{matrix}\right.\\\left[{}\begin{matrix}x=4-2.1=4-2=2\\x=4-\frac{2.20}{9}=-\frac{4}{9}\end{matrix}\right.\end{matrix}\right.\)
Vậy phương trình có 2 nghiệm ( x; y ) = \(\left(2;1\right)\), ( x; y ) = \(\left(-\frac{4}{9};\frac{20}{9}\right)\)
a, Ta có : \(\left\{{}\begin{matrix}2x-y=5\\x^2+xy+y^2=7\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\x^2+x\left(2x-5\right)+\left(2x-5\right)^2=7\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\x^2+2x^2-5x+4x^2-20x+25=7\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\7x^2-25x+18=0\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\7x^2-7x-18x+18=0\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\\left(7x-18\right)\left(x-1\right)=0\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}y=2x-5\\\left[{}\begin{matrix}x=1\\x=\frac{18}{7}\end{matrix}\right.\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}\left[{}\begin{matrix}y=2.1-5=2-5=-3\\y=2.\left(\frac{18}{7}\right)-5=\frac{1}{7}\end{matrix}\right.\\\left[{}\begin{matrix}x=1\\x=\frac{18}{7}\end{matrix}\right.\end{matrix}\right.\)
Vậy hệ phương trình trên có 2 nghiệm là ( x; y ) = ( 1; -3 ) , ( x; y ) \(=\left(\frac{18}{7};\frac{1}{7}\right)\)
