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\(C,\hept{\begin{cases}\left|x-1\right|+\left|y-2\right|=1\\\left|x-1\right|+3y=3\left(#\right)\end{cases}}\)
\(\Rightarrow3y-\left|y-2\right|=2\)(1)
*Nếu y > 2 thì
\(\left(1\right)\Leftrightarrow3y-y+2=2\)
\(\Leftrightarrow y=0\)(Loại do ko tm KĐX)
*Nếu y < 2 thì
\(\left(1\right)\Leftrightarrow3y-2+y=2\)
\(\Leftrightarrow y=1\)(Tm KĐX)
Thay y = 1 vào (#) được \(\left|x-1\right|+3=3\)
\(\Leftrightarrow x=1\)
Vậy hệ có nghiệm \(\hept{\begin{cases}x=1\\y=1\end{cases}}\)
\(A,ĐKXĐ:x\left(y+1\right)>0\)
\(\hept{\begin{cases}x+y=5\left(1\right)\\\sqrt{\frac{x}{y+1}}+\sqrt{\frac{y+1}{x}}=2\left(2\right)\end{cases}}\)
Giải (2)
Có bđt \(\frac{a}{b}+\frac{b}{a}\ge2\left(a,b>0\right)\)
Nên \(\sqrt{\frac{x}{y+1}}+\sqrt{\frac{y+1}{x}}\ge2\)
Dấu "=" xảy ra \(\Leftrightarrow x=y+1\)
Thế x = y + 1 vảo pt (1) được
\(y+1+y=5\)
\(\Leftrightarrow y=2\)
\(\Rightarrow x=2+1=3\)
Thấy x = 3 ; y = 2 thỏa mãn ĐKXĐ
Vậy hệ có ngihiemej \(\hept{\begin{cases}x=3\\y=2\end{cases}}\)
a) \(\hept{\begin{cases}\left(x-1\right)\left(2x+y\right)=0\\\left(y+1\right)\left(2y-x\right)=0\end{cases}}\)
\(\cdot x=1\Rightarrow\hept{\begin{cases}0=0\\\left(y+1\right)\left(2y-1\right)=0\end{cases}}\Leftrightarrow\hept{\begin{cases}0=0\\y=-1;y=\frac{1}{2}\end{cases}}\)
\(\cdot y=-1\Rightarrow\hept{\begin{cases}\left(x-1\right)\left(2x-1\right)=0\\0=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=1;x=\frac{1}{2}\\0=0\end{cases}}\)
\(\cdot x=2y\Rightarrow\hept{\begin{cases}\left(2y-1\right)5y=0\\0=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}y=0\Rightarrow x=0\\y=\frac{1}{2}\Rightarrow x=1\end{cases}}\)
\(y=-2x\Rightarrow\hept{\begin{cases}0=0\\\left(1-2x\right)5x=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{2}\Rightarrow y=-1\\x=0\Rightarrow y=0\end{cases}}\)
b) \(\hept{\begin{cases}x+y=\frac{21}{8}\\\frac{x}{y}+\frac{y}{x}=\frac{37}{6}\end{cases}\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\\left(\frac{21}{8}-y\right)^2+y^2=\frac{37}{6}y\left(\frac{21}{8}-y\right)\end{cases}}}\)
\(\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\2y^2-\frac{21}{4}y+\frac{441}{64}=-\frac{37}{6}y^2+\frac{259}{16}y\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{21}{8}-y\\1568y^2-4116y+1323=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=\frac{3}{8}\\y=\frac{9}{4}\end{cases}}hay\hept{\begin{cases}x=\frac{9}{4}\\y=\frac{3}{8}\end{cases}}\)
c) \(\hept{\begin{cases}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\\\frac{2}{xy}-\frac{1}{z^2}=4\end{cases}\Leftrightarrow\hept{\begin{cases}\frac{1}{z^2}=\left(2-\frac{1}{x}-\frac{1}{y}\right)^2\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}}\)\(\Leftrightarrow\hept{\begin{cases}\left(2xy-x-y\right)^2=-4x^2y^2+2xy\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}8x^2y^2-4x^2y-4xy^2+x^2+y^2-2xy+2xy=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}4x^2y^2-4x^2y+x^2+4x^2y^2-4xy^2+y^2=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\left(2xy-x\right)^2+\left(2xy-y\right)^2=0\\\frac{1}{z^2}=\frac{2}{xy}-4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=y=\frac{1}{2}\\z=\frac{-1}{2}\end{cases}}\)
d) \(\hept{\begin{cases}xy+x+y=71\\x^2y+xy^2=880\end{cases}}\). Đặt \(\hept{\begin{cases}x+y=S\\xy=P\end{cases}}\), ta có: \(\hept{\begin{cases}S+P=71\\SP=880\end{cases}}\Leftrightarrow\hept{\begin{cases}S=71-P\\P\left(71-P\right)=880\end{cases}}\Leftrightarrow\hept{\begin{cases}S=71-P\\P^2-71P+880=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}S=16\\P=55\end{cases}}hay\hept{\begin{cases}S=55\\P=16\end{cases}}\)
\(\cdot\hept{\begin{cases}S=16\\P=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=16\\xy=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x=16-y\\y\left(16-y\right)=55\end{cases}}\Leftrightarrow\hept{\begin{cases}x=16-y\\y^2-16y+55=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x=5\\y=11\end{cases}}hay\hept{\begin{cases}x=11\\y=5\end{cases}}\)
\(\cdot\hept{\begin{cases}S=55\\P=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=55\\xy=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x=55-y\\y\left(55-y\right)=16\end{cases}}\Leftrightarrow\hept{\begin{cases}x=55-y\\y^2-55y+16=0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x=\frac{55-3\sqrt{329}}{2}\\y=\frac{55+3\sqrt{329}}{2}\end{cases}}hay\hept{\begin{cases}x=\frac{55+3\sqrt{329}}{2}\\y=\frac{55-3\sqrt{329}}{2}\end{cases}}\)
e) \(\hept{\begin{cases}x\sqrt{y}+y\sqrt{x}=12\\x\sqrt{x}+y\sqrt{y}=28\end{cases}}\). Đặt \(\hept{\begin{cases}S=\sqrt{x}+\sqrt{y}\\P=\sqrt{xy}\end{cases}}\), ta có \(\hept{\begin{cases}SP=12\\P\left(S^2-2P\right)=28\end{cases}}\Leftrightarrow\hept{\begin{cases}S=\frac{12}{P}\\P\left(\frac{144}{P^2}-2P\right)=28\end{cases}}\Leftrightarrow\hept{\begin{cases}S=\frac{12}{P}\\2P^4+28P^2-144P=0\end{cases}}\)
Tự làm tiếp nhá! Đuối lắm luôn
Ta có \(\hept{\begin{cases}\text{(x+y)(y+z)=187}\\\text{(y+z)(z+x)=154}\\\text{(z+x)(x+y)=238}\end{cases}}\)\(\Rightarrow\)(x+y)2(y+z)2(z+x)2=187.154.238 \(\Rightarrow\) (x+y)(y+z)(z+x)=2618
\(\Rightarrow\)\(\hept{\begin{cases}z+x=14\\x+y=17\\y+z=11\end{cases}}\) \(\Rightarrow\) 2(x+y+z)=14+17+11=42 \(\Rightarrow\) x+y+z=21 \(\Rightarrow\) \(\hept{\begin{cases}y=7\\z=4\\x=10\end{cases}}\)
đặt x+y=a,y+z=b,z+y=c
hPt trở thành :ab=187,bc=154,ca=238
nhân hết 3 vế với nhau:\(a^2b^2c^2=6853924\)
Suy ra \(abc=2613\)nên c=abc:ab=2613:187=14.b và c tính tương tự
trở về ẩn cũ r giải nốt đi
\(Taco:\)
\(\left(x+y\right)\left(y+z\right)=187\Leftrightarrow xy+xz+yy+yz=187\)
\(\left(y+z\right)\left(z+x\right)=154\Leftrightarrow yz+xy+zz+xz=154\)
\(\left(z+x\right)\left(x+y\right)=238\Leftrightarrow xz+zy+xx+xy=238\)
\(\Rightarrow\left(x+y\right)\left(y+z\right)+\left(x+z\right)\left(x+y\right)+\left(y+z\right)\left(z+x\right)=579\)
\(\Leftrightarrow xy+zx+yy+yz+yz+xy+zz+xz+xz+zy+xx+xy=579\)
\(\Leftrightarrow3\left(xz+xy+yz\right)+x^2+y^2+z^2=579\)
\(\left(z+x\right)\left(x+y\right)-\left(x+y\right)\left(y+z\right)=51\)
\(\Leftrightarrow\left(x+y\right)\left(x-y\right)=x^2-y^2=51\)
\(\left(z+x\right)\left(x+y\right)-\left(y+z\right)\left(x+z\right)=84\)
\(\Leftrightarrow\left(x+z\right)\left(x-z\right)=84\Leftrightarrow x^2-z^2=84\)
\(\Leftrightarrow y^2-z^2=33\)
đến đây tịt
a) \(\hept{\begin{cases}\left(x+1\right)\left(y+1\right)=8\\x\left(x+1\right)+y\left(y+1\right)+xy=17\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x+y+xy=7\\x^2+y^2+x+y+xy=17\end{cases}}\)
Dat \(\hept{\begin{cases}xy=P\\x+y=S\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}S+P=7\\S^2+S-P=17\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}P=7-S\\S^2+S-\left(7-S\right)=17\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}P=7-S\\S^2+2S=24\end{cases}}\)
\(\hept{\begin{cases}S=-6\\P=13\\S=4;P=3\end{cases}}\)
b)