2x+3y+4xy=9

<=>4x+8xy+6y+3=21

<=>4x(2y+1)+3(2y+1)=21

<=>(2y+1)(4x+3)=21

đến đây lập bảng xét ước là ra

bạn viết lại đề đi, có số mũ, xuống dòng chứ thế này ai mà giải được

\(xy-2x+3y=13\)

\(x\left(y-2\right)+3y-6=13-6\)

\(x\left(y-2\right)+3\left(y-2\right)=7\)

\(\left(y-2\right)\left(x+3\right)=7\)

\(\Rightarrow\left(y-2\right);\left(x+3\right)\in\text{Ư}\left(7\right)=\left\{\pm1;\pm7\right\}\)

Lập bảng giá trị

Vậy có các cặp số (x;y) là: (-2;9);(-4;-5);(4;3);(-10;1)

Tham khảo nhé~

\(xy-2x+3y=13\Leftrightarrow x\left(y-2\right)+3y-6=7\)

\(\Leftrightarrow x\left(y-2\right)+3\left(y-2\right)=7\Leftrightarrow\left(y-2\right)\left(x+3\right)=7\)

Tự làm tiếp nha !

a) x²+y²-4x+4y+8=0

⇔ (x-2)²+(y+2)²=0

\(\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\y+2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=-2\end{matrix}\right.\)

b)5x²-4xy+y²=0

⇔ x²+(2x-y)²=0

\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\2x-y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)

c)x²+2y²+z²-2xy-2y-4z+5=0

⇔ (x-y)²+(y-1)²+(z-2)²=0

\(\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\y-1=0\\z-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y=1\\z=2\end{matrix}\right.\)

b: Ta có: \(5x^2-4xy+y^2=0\)

\(\Leftrightarrow x^2-\dfrac{4}{5}xy+y^2=0\)

\(\Leftrightarrow x^2-2\cdot x\cdot\dfrac{2}{5}y+\dfrac{4}{25}y^2+\dfrac{21}{25}y^2=0\)

\(\Leftrightarrow\left(x-\dfrac{2}{5}y\right)^2+\dfrac{21}{25}y^2=0\)

Dấu '=' xảy ra khi \(\left\{{}\begin{matrix}x=0\\y=0\end{matrix}\right.\)

d)3x²+3y²+3xy-3x+3y+3=0

⇔ 6x²+6y²+6xy-6x+6y+6=0

⇔ 3(x+y)²+3(x-1)²+3(y+1)²=0

\(\Leftrightarrow\left\{{}\begin{matrix}x+y=0\\x-1=0\\y+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\)

\(3x^2+3y^2+4xy+2x-2y+2=0\)

\(\Leftrightarrow\left(2x^2+2y^2+4xy\right)+\left(x^2+2x+1\right)+\left(y^2-2y+1\right)=0\)

\(\Leftrightarrow2\left(x^2+y^2+2xy\right)+\left(x+1\right)^2+\left(y-1\right)^2=0\)

\(\Leftrightarrow2\left(x+y\right)^2+\left(x+1\right)^2+\left(y-1\right)^2=0\)

Vì \(\left(x+y\right)^2\ge0\); \(\left(x+1\right)^2\ge0\); \(\left(y-1\right)^2\ge0\)\(\forall x,y\)

\(\Rightarrow2\left(x+y\right)^2+\left(x+1\right)^2+\left(y-1\right)^2\ge0\)

Dấu " = " xảy ra \(\Leftrightarrow\hept{\begin{cases}x+y=0\\x+1=0\\y-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-y\\x=-1\\y=1\end{cases}}\)

Vậy \(x=-1\)và \(y=1\)

\(\Leftrightarrow\left(x^2-4xy+4y^2\right)-\left(y^2-4y+4\right)=-1\\ \Leftrightarrow\left(x-2y\right)^2-\left(y-2\right)^2=-1\\ \Leftrightarrow\left(x-2y-y+2\right)\left(x-2y+y-2\right)=-1\\ \Leftrightarrow\left(x-3y+2\right)\left(x-y-2\right)=-1=\left(-1\right)\cdot1\)

\(TH_1:\left\{{}\begin{matrix}x-3y+2=1\\x-y-2=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-3y=-1\\x-y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\\ TH_2:\left\{{}\begin{matrix}x-3y+2=-1\\x-y-2=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-3y=-3\\x-y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=6\\y=3\end{matrix}\right.\)

Vậy PT có nghiệm \(\left(x;y\right)\in\left\{\left(2;1\right);\left(6;3\right)\right\}\)

\(\Leftrightarrow\left(x^2-4xy+4y^2\right)-\left(y^2-4y+4\right)+1=0\\ \Leftrightarrow\left(x-2y^2\right)-\left(y-2\right)^2=-1\\ \Leftrightarrow\left(x-2y-y+2\right)\left(x-2y+y-2\right)=-1\\ \Leftrightarrow\left(x-3y+2\right)\left(x-y-2\right)=-1\)

Vì \(x,y\in Z\Rightarrow\left\{{}\begin{matrix}x-y-2\in Z\\x-3y+2\in Z\\x-y-2,x-3y+2\inƯ\left(-1\right)=\left\{-1;1\right\}\end{matrix}\right.\)

Ta có bảng: