\(\hept{\begin{cases}5|x-1|-3|y+2|=7\\2\sqrt{4x^2-8x+4}+5\sqrt{y^2+4y+4}=13\end{cases}}\)

<=>\(\hept{\begin{cases}5|x-1|-3|y+2|=7\\2\sqrt{4\left(x-1\right)^2}+5\sqrt{\left(y+2\right)^2}=13\end{cases}}\)

<=> \(\hept{\begin{cases}5|x-1|-3|y+2|=7\\4\left|x-1\right|+5\left|y+2\right|=13\end{cases}}\)

<=> \(\hept{\begin{cases}\left|x-1\right|=2\\\left|y+2\right|=1\end{cases}}\)

Giải: |x - 1 | = 2  <=> \(x-1=\pm2\Leftrightarrow\orbr{\begin{cases}x=-1\\x=3\end{cases}}\)

Giải: \(\left|y+2\right|=1\Leftrightarrow\orbr{\begin{cases}y+2=1\\y+2=-1\end{cases}}\Leftrightarrow\orbr{\begin{cases}y=-1\\y=-3\end{cases}}\)

Vậy hệ có tập nghiệm : S = { ( -1; -1) , (-1; -3) ; ( 3; -1) ; (3; -3 )}

\(\hept{\begin{cases}5\left|x-1\right|-3\left|y+2\right|=7\\2\sqrt{4x^2-8x+4}+5\sqrt{y^2+4y+4}=13\end{cases}}\)

<=> \(\hept{\begin{cases}5\left|x-1\right|-3\left|y+2\right|=7\\2\sqrt{\left(2x-2\right)^2}+5\sqrt{\left(y+2\right)^2}=13\end{cases}}\)

<=> \(\hept{\begin{cases}5\left|x-1\right|-3\left|y+2\right|=7\\2\left|2x-2\right|+5\left|y+2\right|=13\end{cases}}\)

<=> \(\hept{\begin{cases}5\left|x-1\right|-3\left|y+2\right|=7\\4\left|x-1\right|+5\left|y+2\right|=13\end{cases}}\)

<=> \(\hept{\begin{cases}\left|x-1\right|=2\\\left|y+2\right|=1\end{cases}}\)

Em tự làm tiếp ( hệ có 4 nghiệm nhé!)

a:

ĐKXĐ: y+1>=0

=>y>=-1

 \(\left\{{}\begin{matrix}2\left(x^2-2x\right)+\sqrt{y+1}=0\\3\left(x^2-2x\right)-2\sqrt{y+1}+7=0\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2\left(x^2-2x\right)+\sqrt{y+1}=0\\3\left(x^2-2x\right)-2\sqrt{y+1}=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}4\left(x^2-2x\right)+2\sqrt{y+1}=0\\3\left(x^2-2x\right)-2\sqrt{y+1}=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}7\left(x^2-2x\right)=-7\\3\left(x^2-2x\right)-2\sqrt{y+1}=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x^2-2x=-1\\3\cdot\left(-1\right)-2\sqrt{y+1}=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x^2-2x+1=0\\2\sqrt{y+1}=-3+7=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(x-1\right)^2=0\\\sqrt{y+1}=2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x-1=0\\y+1=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=3\left(nhận\right)\end{matrix}\right.\)

b: \(\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\2\sqrt{4x^2-8x+4}+5\sqrt{y^2+4y+4}=13\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\2\cdot\sqrt{\left(2x-2\right)^2}+5\cdot\sqrt{\left(y+2\right)^2}=13\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\4\left|x-1\right|+5\left|y+2\right|=13\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}20\left|x-1\right|-12\left|y+2\right|=28\\20\left|x-1\right|+25\left|y+2\right|=65\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-37\left|y+2\right|=-37\\4\left|x-1\right|+5\left|y+2\right|=13\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\left|y+2\right|=1\\4\left|x-1\right|=13-5=8\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left|y+2\right|=1\\\left|x-1\right|=2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x-1\in\left\{2;-2\right\}\\y+2\in\left\{1;-1\right\}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\in\left\{3;-1\right\}\\y\in\left\{-1;-3\right\}\end{matrix}\right.\)

c: ĐKXĐ: \(\left\{{}\begin{matrix}x< >-1\\y< >-4\end{matrix}\right.\)

\(\left\{{}\begin{matrix}\dfrac{3x}{x+1}-\dfrac{2}{y+4}=4\\\dfrac{2x}{x+1}-\dfrac{5}{y+4}=9\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{3x+3-3}{x+1}-\dfrac{2}{y+4}=4\\\dfrac{2x+2-2}{x+1}-\dfrac{5}{y+4}=9\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3-\dfrac{3}{x+1}-\dfrac{2}{y+4}=4\\2-\dfrac{2}{x+1}-\dfrac{5}{y+4}=9\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{3}{x+1}+\dfrac{2}{y+4}=3-4=-1\\\dfrac{2}{x+1}+\dfrac{5}{y+4}=2-9=-7\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{6}{x+1}+\dfrac{4}{y+4}=-2\\\dfrac{6}{x+1}+\dfrac{15}{y+4}=-21\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{-11}{y+4}=19\\\dfrac{3}{x+1}+\dfrac{2}{y+4}=-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y+4=-\dfrac{11}{19}\\\dfrac{3}{x+1}+2:\dfrac{-11}{19}=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{11}{19}-4=-\dfrac{87}{19}\\\dfrac{3}{x+1}=-1-2:\dfrac{-11}{19}=-1+2\cdot\dfrac{19}{11}=\dfrac{27}{11}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=-\dfrac{87}{19}\\x+1=\dfrac{11}{9}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{87}{19}\\x=\dfrac{2}{9}\end{matrix}\right.\)(nhận)

d:

ĐKXĐ: x<>1 và y<>-2

\(\left\{{}\begin{matrix}\dfrac{x+1}{x-1}+\dfrac{3y}{y+2}=7\\\dfrac{2}{x-1}-\dfrac{5}{y+2}=4\end{matrix}\right.\)

=> \(\left\{{}\begin{matrix}\dfrac{x-1+2}{x-1}+\dfrac{3y+6-6}{y+2}=7\\\dfrac{2}{x-1}-\dfrac{5}{y+2}=4\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}1+\dfrac{2}{x-1}+3-\dfrac{6}{y+2}=7\\\dfrac{2}{x-1}-\dfrac{5}{y+2}=4\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{2}{x-1}-\dfrac{6}{y+2}=7-4=3\\\dfrac{2}{x-1}-\dfrac{5}{y+2}=4\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-\dfrac{1}{y+2}=-1\\\dfrac{2}{x-1}-\dfrac{5}{y+2}=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y+2=1\\\dfrac{2}{x-1}-5=4\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=-1\\\dfrac{2}{x-1}=9\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-1\\x-1=\dfrac{2}{9}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-1\\x=\dfrac{11}{9}\end{matrix}\right.\left(nhận\right)\)

\(\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\2\sqrt{4x^2-8x+4}+5\sqrt{y^2+4y+4}=13\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\2\sqrt{\left(2x-2\right)^2}+5\sqrt{\left(y+2\right)^2}=13\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}5\left|x-1\right|-3\left|y+2\right|=7\\2.2\left|x-1\right|+5\left|y+2\right|=13\end{matrix}\right.\\ \left|x-1\right|=a;\left|y+2\right|=b\\ \Rightarrow\left\{{}\begin{matrix}5a-3b=7\\4a+5b=13\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}25a-15b=35\\12a+15b=39\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}37a=74\\5a-3b=7\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}a=2\\5.2-3b=7\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x-1=2\\x-1=-2\end{matrix}\right.\\\left[{}\begin{matrix}y+2=1\\y+2=-1\end{matrix}\right.\end{matrix}\right.\\ \left\{{}\begin{matrix}\left[{}\begin{matrix}x=3\\x=-1\end{matrix}\right.\\\left[{}\begin{matrix}y=-1\\y=-3\end{matrix}\right.\end{matrix}\right.\)

bạn thêm dấu \("\Leftrightarrow"\) ở cuối nhé

a/ ĐKXĐ: \(x\ge4\)

Đặt \(\sqrt{x+4}+\sqrt{x-4}=a>0\)

\(\Rightarrow a^2=2x+2\sqrt{x^2-16}\)

Phương trình trở thành:

\(a=a^2-12\Leftrightarrow a^2-a-12=0\Rightarrow\left[{}\begin{matrix}a=4\\a=-3\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt{x+4}+\sqrt{x-4}=4\)

\(\Leftrightarrow2x+2\sqrt{x^2-16}=16\)

\(\Leftrightarrow\sqrt{x^2-16}=8-x\left(x\le8\right)\)

\(\Leftrightarrow x^2-16=x^2-16x+64\)

\(\Rightarrow x=5\)

b/ \(x\ge-\frac{1}{2}\)

Đặt \(\left\{{}\begin{matrix}\sqrt{2x+1}=a\\\sqrt{4x^2-2x+1}=b\end{matrix}\right.\) ta được:

\(a+3b=3+ab\)

\(\Leftrightarrow ab-a-\left(3b-3\right)=0\)

\(\Leftrightarrow a\left(b-1\right)-3\left(b-1\right)=0\)

\(\Leftrightarrow\left(a-3\right)\left(b-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a=3\\b=1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\sqrt{2x+1}=3\\\sqrt{4x^2-2x+1}=1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}2x+1=9\\4x^2-2x=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=4\\x=0\\x=\frac{1}{2}\end{matrix}\right.\)

Bài 2:

a/ \(\left\{{}\begin{matrix}\left(x+2y\right)^2-4xy-5=0\\4xy\left(x+2y\right)+5\left(x+2y\right)-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+2y\right)^2-\left(4xy+5\right)=0\\\left(4xy+5\right)\left(x+2y\right)-1=0\end{matrix}\right.\)

Đặt \(\left\{{}\begin{matrix}x+2y=a\\4xy+5=b\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a^2-b=0\\ab=1\end{matrix}\right.\) \(\Rightarrow a^2-\frac{1}{a}=0\Rightarrow a^3-1=0\)

\(\Rightarrow\left\{{}\begin{matrix}a=1\\b=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x+2y=1\\4xy+5=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=1-2y\\4y\left(1-2y\right)+4=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=1-2y\\-8y^2+4y+4=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}y=1\Rightarrow x=-1\\y=-\frac{1}{2}\Rightarrow x=2\end{matrix}\right.\)

b/Cộng vế với vế:

\(17x^2-2\left(4y^2+1\right)x+y^4+1=0\)

\(\Delta'=\left(4y^2+1\right)^2-17\left(y^4+1\right)=-y^4+8y^2-16\)

\(\Delta'=-\left(y^2-4\right)^2\ge0\Rightarrow y^2-4=0\Rightarrow\left[{}\begin{matrix}y=2\\y=-2\end{matrix}\right.\)

- Với \(y=2\) \(\Rightarrow x^2-2x+1=0\Rightarrow x=1\)

\(\)- Với \(y=-2\Rightarrow x^2-2x-7=0\Rightarrow x=1\pm2\sqrt{2}\)

b) Ta có pt \(\Leftrightarrow\sqrt{x-1-4\sqrt{x-1}+4}+\sqrt{x-1-6\sqrt{x-1}+9}=1\)

<=>  \(\left|\sqrt{x-1}-2\right|+\left|\sqrt{x-1}-3\right|=1\Leftrightarrow\left|3-\sqrt{x-1}\right|+\left|\sqrt{x-1}-2\right|=1\)

Mà \(\left|3-\sqrt{x-1}\right|+\left|\sqrt{x-1}-2\right|\ge\left|3-\sqrt{x-1}+\sqrt{x-1}-2\right|=1\)

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a) Đặt \(\sqrt{x^2-4x-5}=a\left(a\ge0\right)\)

Ta có pt \(\Leftrightarrow2a^2-3a-2=0\Leftrightarrow\left(a-2\right)\left(2a+1\right)=0\)

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