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a: \(\left\{{}\begin{matrix}3x-2y=1\\2x+4y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6x-4y=2\\2x+4y=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}8x=5\\3x-2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{8}\\2y=3x-1=\dfrac{15}{8}-1=\dfrac{7}{8}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{8}\\y=\dfrac{7}{16}\end{matrix}\right.\)
b: \(\left\{{}\begin{matrix}4x-3y=1\\-x+2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4x-3y=1\\-4x+8y=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=-1+2y=-1+2=1\end{matrix}\right.\)
c: \(\left\{{}\begin{matrix}\dfrac{2}{3}x+\dfrac{4}{3}y=1\\\dfrac{1}{2}x-\dfrac{3}{4}y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+4y=3\\2x-3y=8\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{41}{14}\\y=-\dfrac{5}{7}\end{matrix}\right.\)
Còn lại tương tự
hỏi trước tí, bạn biết giải cái hệ này chứ?
\(\left\{{}\begin{matrix}2x+y=3\\2x-3y=1\end{matrix}\right.\)
1) hpt \(\Leftrightarrow\left\{{}\begin{matrix}x+4y=2\\6x+4y=8\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2-x}{4}\\5x=6\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{1}{5}\\x=\dfrac{6}{5}\end{matrix}\right.\)
Kl: x=6/5 và y=1/5
2) hpt \(\Leftrightarrow\left\{{}\begin{matrix}-2x-2y=4\\-2x-4y=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-2-y\\2y=4\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-4\\y=2\end{matrix}\right.\)
Kl...
3) hpt \(\Leftrightarrow\left\{{}\begin{matrix}2x-3y=2\\2x-3y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{2+3y}{2}\\0=3\left(vô-lý\right)\end{matrix}\right.\)
kl: hpt vn
1/
\(\left\{{}\begin{matrix}3x+2y=6\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+2y=6\\3x-3y=6\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}5y=0\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=0\\x=2\end{matrix}\right.\)
Vậy hệ phương trình đã cho có nghiệm duy nhất \(\left(x;y\right)=\left(2;0\right)\)
2/
\(\left\{{}\begin{matrix}2x-3y=1\\-4x+6y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4x-6y=2\\-4x+6y=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}0x=4\\-4x+6y=2\end{matrix}\right.\)
Vì 0x=4 vô nghiệm \(\Rightarrow-4x+6y=2\) vô nghiệm
Vậy hệ phương trình đã cho vô nghiệm
3/ \(\left\{{}\begin{matrix}2x+3y=5\\5x-4y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}10x+15y=25\\10x-8y=2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}23y=23\\5x-4y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=1\\5x-4=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}y=1\\x=1\end{matrix}\right.\)
Vậy hệ phương trình đã cho có nghiệm duy nhất (x;y) = (1;1)
Giải hệ sau :
Câu a :
\(\left\{{}\begin{matrix}x+y=-1\\2x+y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+y=-1\\-x=-2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=-1\\x=2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}y=-3\\x=2\end{matrix}\right.\)
Vậy ...........................
Câu b :
Đặt \(\left\{{}\begin{matrix}\dfrac{1}{x}=a\\\dfrac{1}{y}=b\end{matrix}\right.\) . Ta có :
\(\left\{{}\begin{matrix}a+b=\dfrac{1}{5}\\3a+4b=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3a+3b=\dfrac{3}{5}\\3a+4b=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-b=-\dfrac{7}{5}\\3a+4b=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b=\dfrac{7}{5}\\a=-\dfrac{6}{5}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{x}=\dfrac{7}{5}\\\dfrac{1}{y}=-\dfrac{6}{5}\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=\dfrac{5}{7}\\y=-\dfrac{5}{6}\end{matrix}\right.\)
Vậy..................
\(a,\left\{{}\begin{matrix}2x-y=4\\x+5y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x-y=4\\2x+10y=6\end{matrix}\right.\left\{{}\begin{matrix}11y=2\\2x+10y=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2}{11}\\2x+10.\dfrac{2}{11}=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2}{11}\\2x=\dfrac{46}{11}\end{matrix}\right.\left\{{}\begin{matrix}y=\dfrac{2}{11}\\x=\dfrac{23}{11}\end{matrix}\right.\)
Câu a)
Áp dụng BĐT AM-GM cho 2 số không âm:
\(\sqrt{\frac{1-x}{2y+1}}+\sqrt{\frac{2y+1}{1-x}}\geq 2\sqrt{\sqrt{\frac{1-x}{2y+1}}.\sqrt{\frac{2y+1}{1-x}}}=2\sqrt{1}=2\)
Dấu "=" xảy ra khi \(\sqrt{\frac{1-x}{2y+1}}=\sqrt{\frac{2y+1}{1-x}}\Rightarrow \frac{1-x}{2y+1}=\frac{2y+1}{1-x}\)
\(\Leftrightarrow \frac{-y}{2y+1}=\frac{2y+1}{-y}\) (do \(x-y=1\) )
\(\Rightarrow y^2=(2y+1)^2\)
\(\Leftrightarrow (2y+1-y)(2y+1+y)=0\Rightarrow \left[\begin{matrix} y=-1\\ y=-\frac{1}{3}\end{matrix}\right.\)
Thử lại thấy chỉ \(y=-\frac{1}{3}\) thỏa mãn kéo theo \(x=1+y=\frac{2}{3}\)
Vậy \((x,y)=(\frac{2}{3}; \frac{-1}{3})\)
Câu b)
Thay \(y=2x-1\) vào pt thứ nhất ta có:
\(|x-(2x-1)|=|2(2x-1)-1|\)
\(\Leftrightarrow |1-x|=|4x-3|\)
\(\Rightarrow \left[\begin{matrix} 1-x=4x-3\\ 1-x=3-4x\end{matrix}\right.\Rightarrow \left[\begin{matrix} x=\frac{4}{5}\\ x=\frac{2}{3}\end{matrix}\right.\)
\(\Rightarrow \left[\begin{matrix} y=\frac{3}{5}\\ y=\frac{1}{3}\end{matrix}\right.\) (tương ứng)
\(\left\{{}\begin{matrix}x+2y=8\\\dfrac{1}{2}x-y=18\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x+2y=8\\x-2y=36\end{matrix}\right.\)
\(\left\{{}\begin{matrix}2x=44\\x+2y=8\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x=22\\y=-7\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x+2y=8\\\dfrac{1}{2}x-y=18\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x+2y=8\\x-2y=36\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}2x=44\\x-2y=36\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=22\\2y=x-36=22-36=-14\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x=22\\x=-\dfrac{14}{2}=-7\end{matrix}\right.\)
Vậy: ...