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\(\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}=5\\\dfrac{1}{xy}=6\end{matrix}\right.\left(x,y\ne0\right)\)\(\Leftrightarrow\left(I\right)\left\{{}\begin{matrix}\dfrac{1}{x}+\dfrac{1}{y}=5\\\dfrac{1}{x}\cdot\dfrac{1}{y}=6\end{matrix}\right.\)
Đặt \(\dfrac{1}{x}=a,\dfrac{1}{y}=b\left(a,b>0\right)\)
Hệ (I) \(\Leftrightarrow\left\{{}\begin{matrix}a+b=5\\ab=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\ab=6\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\b\left(5-b\right)=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\-\left(b^2-5b\right)=6\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\b^2-5b+6=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\\left(b-3\right)\left(b-2\right)=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=5-b\\b-3=0\end{matrix}\right.\\\left\{{}\begin{matrix}a=5-b\\b-2=0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=5-b\\b=3\end{matrix}\right.\\\left\{{}\begin{matrix}a=5-b\\b=2\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=2\\b=3\end{matrix}\right.\\\left\{{}\begin{matrix}a=3\\b=2\end{matrix}\right.\end{matrix}\right.\)
Trả lại biến cũ
\(\left\{{}\begin{matrix}a=2\\b=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{x}=2\\\dfrac{1}{y}=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0,5\\y=\dfrac{1}{3}\end{matrix}\right.\left(TM\right)\)và ngược lại
Vậy HPT có các cặp nghiệm là \(\left(0,5;\dfrac{1}{3}\right);\left(\dfrac{1}{3};0,5\right)\)
P/S: Bạn kiểm tra kết quả lại giúp mình nhé
\(\left\{{}\begin{matrix}3x+y=3\\3x-y=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6x=0\\3x+y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=3\end{matrix}\right.\)
\(\left\{{}\begin{matrix}3x+y=3\\3x-y=-3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}3x+y+3x-y=3-3\\3x-y=-3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}6x=0\\3x-y=-3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\3.0-y=-3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=3\end{matrix}\right.\)
=>12/(x+y-1)-15/(2x-y+3)=15/2 và 12/(x+y-1)-4/(2x-y+3)=28/5
=>x+y-1=22/9; 2x-y+3=-110/19
=>x+y=31/9; 2x-y=-167/19
=>x=-914/513; y=2681/513
\(\left\{{}\begin{matrix}\dfrac{4}{x+y-1}-\dfrac{5}{2x-y+3}=\dfrac{5}{2}\\\dfrac{3}{x+y-1}-\dfrac{1}{2x-y+3}=\dfrac{7}{5}\end{matrix}\right.\)
a) Thay m=1 vào hệ phương trình, ta được:
\(\left\{{}\begin{matrix}x+2y=2\\2x+3y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+4y=4\\2x+3y=-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=5\\x+2y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=5\\x+10=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-8\\y=5\end{matrix}\right.\)
Vậy: Khi m=1 thì hệ phương trình có nghiệm duy nhất là (x,y)=(-8;5)
b) Ta có: \(\left\{{}\begin{matrix}x+2y=m+1\\2x+3y=m-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+4y=2m+2\\2x+3y=m-2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=m+4\\x+2\cdot\left(m+4\right)=m+1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+2m+8=m+1\\y=m+4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=-m-7\\y=m+4\end{matrix}\right.\)
Để hệ phương trình có nghiệm (x,y) thỏa mãn x>3 và y<5 thì \(\left\{{}\begin{matrix}-m-7>3\\m+4< 5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-m>10\\m< 1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m< -10\\m< 1\end{matrix}\right.\Leftrightarrow m< -10\)
Vậy: Để hệ phương trình có nghiệm (x,y) thỏa mãn x>3 và y<5 thì m<-10
HPT\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2-xy=1-2xy\\\left(x+y\right)\left(1-2xy\right)=x+3y\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2+xy=1\\x^2+xy=-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2+xy=1\\y=-\sqrt{2};\sqrt{2}\end{matrix}\right.\)
The vao roi tinh la xong
\(\left\{{}\begin{matrix}x^3+y^3=^{ }1\left(1\right)\\x^5+y^5=x^2+y^2\left(2\right)\end{matrix}\right.\)
(2)\(\Leftrightarrow x^5-x^2+y^5-y^2=0\)
\(\Leftrightarrow x^2\left(x^3-1\right)+y^2\left(y^3-1\right)=0\)
\(\Leftrightarrow x^2\left(-y\right)^3+y^2\left(-x\right)^3=0\)
\(\Leftrightarrow x^2y^3+y^2x^3=0\)
\(\Leftrightarrow x^2y^2\left(x+y\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\Rightarrow y=1\\y=0\Rightarrow x=1\\x=-y\left(loại\right)\end{matrix}\right.\)
\(\left\{{}\begin{matrix}4x+3x=-6\\\dfrac{x+3y}{3}-\dfrac{y-2}{5}=1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}7x=-6\\\dfrac{5\left(x+3y\right)-3\left(y-2\right)}{15}=1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=-\dfrac{6}{7}\\5x+15y-3y+6=15\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=-\dfrac{6}{7}\\12y=9-5x=9+5\cdot\dfrac{6}{7}=9+\dfrac{30}{7}=\dfrac{93}{7}\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=-\dfrac{6}{7}\\y=\dfrac{93}{7\cdot12}=\dfrac{93}{84}=\dfrac{31}{28}\end{matrix}\right.\)