Giải hệ phương trình: \(\left\{{}\begin{matrix}\sqrt{\frac{x+3}{x}}+\frac{2y}{y-2}=8\\2\sqrt{\frac{x+3}{x}}+\frac{3y}{y-2}=13\end{matrix}\right.\)
giải hệ phương trình:
1, \(\left\{{}\begin{matrix}2y\left(4y^2+3x^2\right)=x^4\left(x^2+3\right)\\2012^x\left(\sqrt{2y-2x+5}-x+1\right)=4024\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\\2x+\frac{1}{x+y}=1\end{matrix}\right.\)
\(2,\left\{{}\begin{matrix}x^3-2x^2y-15x=6y\left(2x-5-4y\right)\left(1\right)\\\frac{x^2}{8y}+\frac{2x}{3}=\sqrt{\frac{x^3}{3y}+\frac{x^2}{4}}-\frac{y}{2}\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\left(2y-x\right)\left(x^2-12y-15\right)=0\)\(\Leftrightarrow\left[{}\begin{matrix}2y=x\\y=\frac{x^2-15}{12}\end{matrix}\right.\)
Ta xét các trường hợp sau:
Trường hợp 1:
\(y=\frac{x^2-15}{12}\) thay vào phương trình \(\left(2\right)\) ta được:
\(\frac{3x^2}{2\left(x^2-15\right)}+\frac{2x}{3}=\sqrt{\frac{4x^3}{x^2-15}+\frac{x^2}{4}}-\frac{x^2-15}{24}\)
\(\Leftrightarrow\frac{36x^2}{x^2-15}-12\sqrt{\frac{x^2}{x^2-15}\left(x^2+16x-15\right)}+\left(x^2+16x-15\right)=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\6\sqrt{\frac{x^2}{x^2-15}}=\sqrt{\left(x^2+16x-15\right)}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\36\frac{x^2}{x^2-15}=x^2+16x-15\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2+16x-15\ge0\\36x^2=\left(x^2-15\right)\left(x^2+16x-15\right)\left(3\right)\end{matrix}\right.\)
Ta xét phương trình \(\left(3\right):36x^2=\left(x^2-15\right)\left(x^2+16x-15\right)\)
Vì: \(x=0\) Không phải là nghiệm. Ta chia cả hai vế p.trình cho \(x^2\) ta được:
\(36=\left(x-\frac{15}{x}\right)\left(x+16-\frac{15}{x}\right)\)
Đặt: \(x-\frac{15}{x}=t\Rightarrow t^2+16t-36=0\Leftrightarrow\left[{}\begin{matrix}t=2\\t=-18\end{matrix}\right.\)
+ Nếu như:
\(t=2\Leftrightarrow x-\frac{15}{x}=2\Leftrightarrow x^2-2x-15=0\Leftrightarrow\left[{}\begin{matrix}x=5\\x=-3\end{matrix}\right.\)\(\Leftrightarrow x=5\)
+ Nếu như:
\(t=-18\Leftrightarrow x-\frac{15}{x}=-18\Leftrightarrow x^2+18x-15=0\Leftrightarrow\left[{}\begin{matrix}x=-9-4\sqrt{6}\\x=-9+4\sqrt{6}\end{matrix}\right.\Leftrightarrow x=-9-4\sqrt{6}\)
Trường hợp 2:
\(x=2y\) thay vào p.trình \(\left(2\right)\) ta được:
\(\Leftrightarrow\frac{x^2}{4x}+\frac{2x}{3}=\sqrt{\frac{2x^3}{3x}+\frac{x^2}{4}}-\frac{x}{4}\Leftrightarrow\frac{7}{6}x=\sqrt{\frac{11x^2}{12}}\Leftrightarrow x=0\left(ktmđk\right)\)
Vậy nghiệm của hệ đã cho là: \(\left(x,y\right)=\left(5;\frac{5}{6}\right),\left(-9-4\sqrt{6};\frac{27+12\sqrt{6}}{2}\right)\)
Năm mới chắc bị lag @@ tớ sửa luôn đề câu 3 nhé :v
3, \(\left\{{}\begin{matrix}8\left(x^2+y^2\right)+4xy+\frac{5}{\left(x+y\right)^2}=13\left(1\right)\\2xy+\frac{1}{x+y}=1\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow8\left[\left(x+y\right)^2-2xy\right]+4xy+\frac{5}{\left(x+y\right)^2}=13\)
Đặt \(\left\{{}\begin{matrix}x+y=a\\xy=b\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow8\left(a^2-2b\right)+4b+\frac{5}{a^2}=13\)
\(\Leftrightarrow8a^2-12b+\frac{5}{a^2}=13\)
Ta cũng có \(\left(2\right)\Leftrightarrow2b+\frac{1}{a}=1\)
\(\Leftrightarrow2b=1-\frac{1}{a}\)
Thay vào (1) ta được :
\(8a^2+\frac{5}{a^2}-6\cdot\left(1-\frac{1}{a}\right)=13\)
\(\Leftrightarrow8a^2+\frac{5}{a^2}-6+\frac{6}{a}=13\)
\(\Leftrightarrow8a^2+\frac{5}{a^2}+\frac{6}{a}=19\)
Giải pt được \(a=1\)
Khi đó \(b=\frac{1-\frac{1}{1}}{2}=0\)
Ta có hệ :
\(\left\{{}\begin{matrix}x+y=1\\xy=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=0\\y=1\end{matrix}\right.\\\left\{{}\begin{matrix}x=1\\y=0\end{matrix}\right.\end{matrix}\right.\)
Vậy...
1.Giải hệ phương trình:
a.\(\left\{{}\begin{matrix}2\sqrt{2}x+y=2\sqrt{2}\\7x-3y=7\end{matrix}\right.\)
b.\(\left\{{}\begin{matrix}7x+y=-\frac{1}{7}\\-\frac{4}{3}x-2y=1\frac{1}{3}\end{matrix}\right.\)
c.\(\left\{{}\begin{matrix}2\sqrt{5}x+3y=\sqrt{2}\\\sqrt{5}x-y=3\sqrt{2}\end{matrix}\right.\)
d.\(\left\{{}\begin{matrix}\frac{2}{x}+\frac{3}{y}=-5\\\frac{3}{x}-\frac{4}{y}=1\end{matrix}\right.\)
e.\(\left\{{}\begin{matrix}-\frac{5}{3x+1}+\frac{7}{2x+1}=\frac{5}{7}\\\frac{1}{3x+1}-\frac{1}{2y-3}=\frac{2}{7}\\\end{matrix}\right.\)
g.\(\left\{{}\begin{matrix}2x^2+5y^2=129\\-3x^2+y^2=13\end{matrix}\right.\)
Giải hệ phương trình :
1, \(\left\{{}\begin{matrix}x-2y=1\\2x-y=4\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}\frac{x}{y}-\frac{y}{y+12}=1\\\frac{x}{y+12}-\frac{x}{y}=2\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}3x^2+y^2=5\\x^2-3y=1\end{matrix}\right.\)
4, \(\left\{{}\begin{matrix}\sqrt{3x-1}-\sqrt{2y+1}=1\\2\sqrt{3x-1}+3\sqrt{2y+1}=12\end{matrix}\right.\)
a/ Bạn tự giải
b/ ĐKXĐ:...
Cộng vế với vế: \(\frac{x-y}{y+12}=3\Rightarrow x-y=3y+36\Rightarrow x=4y+36\)
Thay vào pt đầu: \(\frac{4y+36}{y}-\frac{y}{y+12}=1\)
Đặt \(\frac{y+12}{y}=a\Rightarrow4a-\frac{1}{a}=1\Rightarrow4a^2-a-1=0\)
\(\Rightarrow a=\frac{1\pm\sqrt{17}}{8}\) \(\Rightarrow\frac{y+12}{y}=\frac{1\pm\sqrt{17}}{8}\)
\(\Rightarrow\left[{}\begin{matrix}y+12=y\left(\frac{1+\sqrt{17}}{8}\right)\\y+12=y\left(\frac{1-\sqrt{17}}{8}\right)\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\left(\frac{-7+\sqrt{17}}{8}\right)y=12\\\left(\frac{-7-\sqrt{17}}{8}\right)y=12\end{matrix}\right.\) \(\Rightarrow y=...\)
Chắc bạn ghi sai đề, nghiệm quá xấu
3/ \(\Leftrightarrow\left\{{}\begin{matrix}3x^2+y^2=5\\3x^2-9y=3\end{matrix}\right.\) \(\Rightarrow y^2+9y=2\Rightarrow y^2+9y-2=0\Rightarrow y=...\)
4/ ĐKXĐ:...
\(\Leftrightarrow\left\{{}\begin{matrix}3\sqrt{3x-1}-3\sqrt{2y+1}=3\\2\sqrt{3x-1}+3\sqrt{2y+1}=12\end{matrix}\right.\)
\(\Rightarrow5\sqrt{3x-1}=15\Rightarrow\sqrt{3x-1}=3\Rightarrow x=\frac{10}{3}\)
\(\sqrt{2y+1}=\sqrt{3x-1}-1=3-1=2\Rightarrow2y+1=4\Rightarrow y=\frac{3}{2}\)
Giải hệ phương trình
a, \(\left\{{}\begin{matrix}\sqrt[4]{x^3-1}+\sqrt{x}=3\\x^2+y^3=82\end{matrix}\right.\) d, \(\left\{{}\begin{matrix}x-\frac{1}{x}=y-\frac{1}{y}\\2y=x^3+1\end{matrix}\right.\)
b, \(\left\{{}\begin{matrix}\sqrt{x+\frac{1}{y}}+\sqrt{x+y-3}=3\\2x+y+\frac{1}{y}=8\end{matrix}\right.\)
c,\(\left\{{}\begin{matrix}\frac{3}{x^2}=2x+y\\\frac{3}{y^2}=2y+x\end{matrix}\right.\)
Bài 2:
ĐK: ..........
Đặt $\sqrt{x+\frac{1}{y}}=a; \sqrt{x+y-3}=b$ $(a,b\geq 0$)
HPT \(\Leftrightarrow \left\{\begin{matrix} a+b=3\\ a^2+b^2+3=8\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a+b=3\\ a^2+b^2=5\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} a+b=3\\ (a+b)^2-2ab=5\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a+b=3\\ ab=2\end{matrix}\right.\)
Áp dụng định lý Vi-et đảo thì $a,b$ là nghiệm của pt $X^2-3X+2=0$
$\Rightarrow (a,b)=(2,1); (1,2)$
Nếu $(a,b)=(2,1)$
\(\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=4\\ x+y-3=1\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=4\\ x+y=4\end{matrix}\right.\Rightarrow y=\frac{1}{y}\Rightarrow y=\pm 1\)
$y=1\rightarrow x=3$
$y=-1\rightarrow y=5$
Nếu $(a,b)=(1,2)$
\(\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=1\\ x+y-3=4\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=1\\ x+y=7\end{matrix}\right.\Rightarrow y-\frac{1}{y}=6\)
\(\Rightarrow y^2-6y-1=0\Rightarrow y=3\pm \sqrt{10}\)
Nếu $y=3+\sqrt{10}\rightarrow x=4-\sqrt{10}$
Nếu $y=3-\sqrt{10}\rightarrow x=4+\sqrt{10}$
Vậy...........
Bài 1:
Đặt $\sqrt[4]{y^3-1}=a; \sqrt{x}=b$ $(a,b\geq 0$)
Khi đó hệ PT trở thành:
\(\left\{\begin{matrix} a+b=3\\ b^4+a^4+1=82\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a+b=3\\ a^4+b^4=81\end{matrix}\right.\)
Có: \(a^4+b^4=81\)
\(\Leftrightarrow (a^2+b^2)^2-2a^2b^2=81\)
\(\Leftrightarrow [(a+b)^2-2ab]^2-2a^2b^2=81\)
\(\Leftrightarrow (9-2ab)^2-2a^2b^2=81\)
\(\Leftrightarrow 2a^2b^2-36ab=0\)
\(\Leftrightarrow ab(ab-18)=0\Rightarrow \left[\begin{matrix} ab=0\\ ab=18\end{matrix}\right.\)
Nếu $ab=0$. Kết hợp với $a+b=3$ suy ra $(a,b)=(3,0); (0,3)$
$\Rightarrow (x,y)=(0, \sqrt[4]{82}); (9, 1)$
Nếu $ab=18$. Kết hợp với $a+b=3$ và định lý Vi-et đảo suy ra $a,b$ là nghiệm của pt: $X^2-3X+18=0$
Dễ thấy pt này vô nghiệm nên loại
Vậy......
Bài 2:
ĐK: ..........
Đặt $\sqrt{x+\frac{1}{y}}=a; \sqrt{x+y-3}=b$ $(a,b\geq 0$)
HPT \(\Leftrightarrow \left\{\begin{matrix} a+b=3\\ a^2+b^2+3=8\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a+b=3\\ a^2+b^2=5\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} a+b=3\\ (a+b)^2-2ab=5\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a+b=3\\ ab=2\end{matrix}\right.\)
Áp dụng định lý Vi-et đảo thì $a,b$ là nghiệm của pt $X^2-3X+2=0$
$\Rightarrow (a,b)=(2,1); (1,2)$
Nếu $(a,b)=(2,1)$
\(\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=4\\ x+y-3=1\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=4\\ x+y=4\end{matrix}\right.\Rightarrow y=\frac{1}{y}\Rightarrow y=\pm 1\)
$y=1\rightarrow x=3$
$y=-1\rightarrow y=5$
Nếu $(a,b)=(1,2)$
\(\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=1\\ x+y-3=4\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x+\frac{1}{y}=1\\ x+y=7\end{matrix}\right.\Rightarrow y-\frac{1}{y}=6\)
\(\Rightarrow y^2-6y-1=0\Rightarrow y=3\pm \sqrt{10}\)
Nếu $y=3+\sqrt{10}\rightarrow x=4-\sqrt{10}$
Nếu $y=3-\sqrt{10}\rightarrow x=4+\sqrt{10}$
Vậy...........
hệ phương trình
1 ,\(\left\{{}\begin{matrix}\frac{2x-3}{2y-5}=\frac{3x+1}{3y-4}\\2\left(x-3\right)-3\left(y+2\right)=-16\end{matrix}\right.\)
2, \(\left\{{}\begin{matrix}\frac{x}{y}=\frac{3}{2}\\3x-2y=5\end{matrix}\right.\)
3, \(\left\{{}\begin{matrix}\frac{x^2-y-6}{x}=x-2\\x+3y=8\end{matrix}\right.\)
4, \(\left\{{}\begin{matrix}\frac{x}{y}=\frac{2}{3}\\x+y=10\end{matrix}\right.\)
5, \(\left\{{}\begin{matrix}\frac{y^2+2x-8}{y}=y-3\\x+y=10\end{matrix}\right.\)
6 , \(\left\{{}\begin{matrix}\frac{x+1}{y-1}=5\\3\left(2x-2\right)-4\left(3x+4\right)=5\end{matrix}\right.\)
7, \(\left\{{}\begin{matrix}2x+y=4\\\left|x-2y\right|=3\end{matrix}\right.\)
8 , \(\left\{{}\begin{matrix}\frac{2x}{x+1}+\frac{y}{y+1}=3\\\frac{x}{x+1}-\frac{3y}{y+1}=-1\end{matrix}\right.\)
9 , \(\left\{{}\begin{matrix}y-\left|x\right|=1\\2x-y=1\end{matrix}\right.\)
10 , \(\left\{{}\begin{matrix}\sqrt{x+3y}=\sqrt{3x-1}\\5x-y=9\end{matrix}\right.\)
Giải hệ pt:
a)\(\left\{{}\begin{matrix}x^2+y^2+x+y=18\\x\left(x+1\right).y\left(y+1\right)=72\end{matrix}\right.\) b) \(\left\{{}\begin{matrix}\frac{1}{x}+\frac{1}{y+1}=1\\3y-1=xy\end{matrix}\right.\) c)\(\left\{{}\begin{matrix}2x+3y=xy+5\\\frac{1}{x}+\frac{1}{y+1}=1\end{matrix}\right.\)
d)\(\left\{{}\begin{matrix}\sqrt{\frac{x}{y}}-3\sqrt{\frac{y}{x}}=2\\x-y+xy=1\end{matrix}\right.\) e)\(\left\{{}\begin{matrix}xy+x+y=x^2-2y^2\\x\sqrt{2y}-y\sqrt{x-1}=2x-2y\end{matrix}\right.\)
HELP ME :((
a/ \(\left\{{}\begin{matrix}\left(x^2+x\right)+\left(y^2+y\right)=18\\\left(x^2+x\right)\left(y^2+y\right)=72\end{matrix}\right.\)
Theo Viet đảo, \(x^2+x\) và \(y^2+y\) là nghiệm của:
\(t^2-18t+72=0\Rightarrow\left[{}\begin{matrix}t=12\\t=6\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x^2+x=6\\y^2+y=12\end{matrix}\right.\\\left\{{}\begin{matrix}x^2+x=12\\y^2+y=6\end{matrix}\right.\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=\left\{2;-3\right\}\\y=\left\{3;-4\right\}\end{matrix}\right.\\\left\{{}\begin{matrix}x=\left\{3;-4\right\}\\y=\left\{2;-3\right\}\end{matrix}\right.\end{matrix}\right.\)
b/ ĐKXĐ: ...
\(\left\{{}\begin{matrix}\frac{1}{x}+\frac{1}{y+1}=1\\x=\frac{3y-1}{y}\end{matrix}\right.\)
Nhận thấy \(y=\frac{1}{3}\) không phải nghiệm
\(\Rightarrow\left\{{}\begin{matrix}\frac{1}{x}+\frac{1}{y+1}=1\\\frac{1}{x}=\frac{y}{3y-1}\end{matrix}\right.\) \(\Rightarrow\frac{y}{3y-1}+\frac{1}{y+1}=1\)
\(\Leftrightarrow y\left(y+1\right)+3y-1=\left(3y-1\right)\left(y+1\right)\)
\(\Leftrightarrow y^2-y=0\Rightarrow\left[{}\begin{matrix}y=0\left(l\right)\\y=1\end{matrix}\right.\) \(\Rightarrow x=2\)
c/ ĐKXĐ: ...
\(\Leftrightarrow\left\{{}\begin{matrix}2x+3y=xy+5\\y+1=xy\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x+2y-1=5\\y+1=xy\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=3\\y+1=xy\end{matrix}\right.\)
\(\Rightarrow y+1=\left(3-y\right)y\)
\(\Leftrightarrow y^2-2y+1=0\Rightarrow y=1\Rightarrow x=2\)
Giải hệ phương trình:
\(\left\{{}\begin{matrix}x+y=800\\\frac{115}{100}x+\frac{112}{100}y=945\end{matrix}\right.\)
\(\left\{{}\begin{matrix}\left(\sqrt{2}+\sqrt{3}\right)x-y\sqrt{2}=\sqrt{2}\\\left(\sqrt{2}+\sqrt{3}\right)x+y\sqrt{3}=-\sqrt{3}\end{matrix}\right.\)
\(\left\{{}\begin{matrix}\frac{3}{x}+y=5\\\frac{2}{x}-2y=-2\end{matrix}\right.\)
Mn giải chi tiết giúp mình nhé!
Giải các hệ phương trình:
\(a,\left\{{}\begin{matrix}x-y=12\\\frac{x}{\sqrt{5}}=\frac{y}{\sqrt{3}}\end{matrix}\right.\)
\(b,\left\{{}\begin{matrix}\frac{x}{2}+3y=\frac{\sqrt{2}}{2}\\x+6y=\sqrt{2}\end{matrix}\right.\)