Đặt \(x+\dfrac{1}{x}=a;y+\dfrac{1}{y}=b\left(\left|a\right|\ge2;\left|b\right|\ge2\right)\)

\(\left\{{}\begin{matrix}x+\dfrac{1}{x}+y+\dfrac{1}{y}=5\\x^3+y^3+\dfrac{1}{x^3}+\dfrac{1}{y^3}=15m-25\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{x}+y+\dfrac{1}{y}=5\\\left(x^3+\dfrac{1}{x^3}\right)+\left(y^3+\dfrac{1}{y^3}\right)=15m-25\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{x}+y+\dfrac{1}{y}=5\\\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)+\left(y+\dfrac{1}{y}\right)^3-3\left(y+\dfrac{1}{y}\right)=15m-25\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{x}+y+\dfrac{1}{y}=5\\\left(x+\dfrac{1}{x}\right)^3+\left(y+\dfrac{1}{y}\right)^3-3\left(x+\dfrac{1}{x}+y+\dfrac{1}{y}\right)=15m-25\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{x}+y+\dfrac{1}{y}=5\\\left(x+\dfrac{1}{x}\right)^3+\left(y+\dfrac{1}{y}\right)^3=15m-10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=5\\a^3+b^3=15m-10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=5\\\left(a+b\right)^3-3ab\left(a+b\right)=15m-10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=5\\125-15ab=15m-10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=5\\ab=9-m\end{matrix}\right.\)

\(\Rightarrow a,b\) là nghiệm của phương trình \(t^2-5t+9-m=0\left(1\right)\)

a, Nếu \(m=3\), phương trình \(\left(1\right)\) trở thành

\(t^2-5t+6=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=2\\t=3\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=2\\b=3\end{matrix}\right.\\\left\{{}\begin{matrix}a=3\\b=2\end{matrix}\right.\end{matrix}\right.\)

TH1: \(\left\{{}\begin{matrix}x+\dfrac{1}{x}=2\\y+\dfrac{1}{y}=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(x-1\right)^2=0\\y^2-3y+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{3\pm\sqrt{5}}{2}\end{matrix}\right.\)

TH2: \(\left\{{}\begin{matrix}x+\dfrac{1}{x}=3\\y+\dfrac{1}{y}=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{3\pm\sqrt{5}}{2}\\y=1\end{matrix}\right.\)

Vậy ...

b, \(\left(1\right)\Leftrightarrow t=\dfrac{5\pm\sqrt{4m-11}}{2}\left(m\ge\dfrac{11}{4}\right)\)

\(\left(1\right)\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{5\pm\sqrt{4m-11}}{2}\\b=\dfrac{5\mp\sqrt{4m-11}}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{x}=\dfrac{5\pm\sqrt{4m-11}}{2}\\y+\dfrac{1}{y}=\dfrac{5\mp\sqrt{4m-11}}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x^2-\left(5\pm\sqrt{4m-11}\right)+2=0\left(2\right)\\2y^2-\left(5\mp\sqrt{4m-11}\right)+2=0\end{matrix}\right.\)

Yêu cầu bài toán thỏa mãn khi phương trình \(\left(2\right)\) có nghiệm dương

\(\Leftrightarrow\left\{{}\begin{matrix}\Delta=\left(5\pm\sqrt{4m-11}\right)^2-16\ge0\\\dfrac{5\pm\sqrt{4m-11}}{2}>0\\1>0\end{matrix}\right.\)

\(\Leftrightarrow...\)

a Để hpt có nghiệm \(\left(x;y\right)=\left(-2;3\right)\) \(\Rightarrow\left\{{}\begin{matrix}-2+3m=4\\-2n+3=-3\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}3m=6\\-2n=-6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m=2\\n=2\end{matrix}\right.\)

b Để hpt có vô số nghiệm \(\Leftrightarrow\dfrac{1}{n}=\dfrac{m}{1}=\dfrac{4}{-3}\) \(\left(\dfrac{a}{a'}=\dfrac{b}{b'}=\dfrac{c}{c'}\right)\) 

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{n}=-\dfrac{4}{3}\\m=-\dfrac{4}{3}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m=-\dfrac{4}{3}\\n=-\dfrac{3}{4}\end{matrix}\right.\)

Vậy...

Đáp án: D

\(\hept{\begin{cases}3x-2y=1\\mx+3y=4\end{cases}}\)

\(\hept{\begin{cases}3x=1+2y\\mx+3y=4\end{cases}}\)

\(\hept{\begin{cases}x=1+\frac{2y}{3}\\mx+3y=4\end{cases}}\)

a, Khi thay m = 1 thì biểu thức mx + 3y ta đc

\(x+3y=4\)

Hệ phương trình trở thành : \(\hept{\begin{cases}x=1+\frac{2y}{3}\\x+3y=4\end{cases}}\)

Ta thay x vào biểu thức x + 3y = 4 ta đc

\(1+\frac{2y}{3}+3y=4\)

\(1+\frac{2y}{3}+\frac{9y}{3}-4=0\)

\(-3+\frac{11y}{3}=0\)

\(\frac{11y}{3}=3\Leftrightarrow11y=9\Leftrightarrow y=\frac{9}{11}\)

Ta thay y = 9/11 vào biểu thức x + 3y ta đc

\(x+3.\frac{9}{11}=4\)

\(x+\frac{27}{11}=4\)

\(x=\frac{17}{11}\)

Vậy \(\left\{x;y\right\}=\left\{\frac{17}{11};\frac{9}{11}\right\}\)

Thay \(a=-\sqrt{2}\) vào pt :

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

Lấy \(\left(1\right)+\left(2\right):\) 

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

\(\Leftrightarrow x=\dfrac{3-\sqrt{2}}{1-2\sqrt{2}}\)

\(\Leftrightarrow x=\dfrac{1-5\sqrt{2}}{7}\)\(\left(3\right)\)

Thay \(\left(3\right)\) vào \(\left(2\right)\) : \(-\sqrt{2}.\dfrac{1-5\sqrt{2}}{7}+y=-\sqrt{2}\)

\(\Rightarrow y=\)\(-\sqrt{2}+\dfrac{6\sqrt{2}}{7}\)

\(\Rightarrow y=-\dfrac{\sqrt{2}}{7}\)

Vậy hệ pt có nghiệm duy nhất \(\left(x;y\right)=\left(\dfrac{1-5\sqrt{2}}{7};-\dfrac{\sqrt{2}}{7}\right)\)

Với m =1 suy ra : 

\(\hept{\begin{cases}2x-y=1\\-x+y=2\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}y=2x-1\\-x+2x-1=2\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}y=2.3-1=5\\x=3\end{cases}}\)

b ) Để hệ có nghiệm x+2y=3 

\(\Rightarrow\hept{\begin{cases}x+2y=3\\-x+y=2\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}x=3-2y\\-\left(3-2y\right)+y=2\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}x=3-2.\frac{5}{3}=-\frac{1}{3}\\y=\frac{5}{3}\end{cases}}\)

\(\Rightarrow2.\left(-\frac{1}{3}\right)-\frac{5}{3}=2m-1\Rightarrow m=-\frac{2}{3}\)

Từ hệ được x+y=1

a)Thay vào được x=1;y=0

b)Với mọi a

c)Thay vào x+y=1 tìm x;y

Thay ngược vào hệ tìm a

a) Khi a = 2 hệ phương trình đã cho tương đương với:

 \(\hept{\begin{cases}x+2x=3\left(1\right)\\2x-y=2\left(2\right)\end{cases}}\Leftrightarrow\hept{\begin{cases}3x=3\\2x-y=2\end{cases}}\Leftrightarrow\hept{\begin{cases}x=1\\2x-2=y\end{cases}}\Leftrightarrow\hept{\begin{cases}x=1\\2.1-2=0=y\end{cases}}\)

Do vậy \(\left(x;y\right)=\left(1;0\right)\)

b) Ta có:  \(x+y=\left(x+ax\right)-\left(ax-y\right)=3-2=1>0\forall a\)

c) Lấy (1) trừ (2),vế với vế,ta có: \(x+y=1\)

Thay vào,ta có: \(\sqrt{2}.y+y=1\Leftrightarrow y\left(\sqrt{2}+1\right)=1\)

\(\Rightarrow y=\frac{1}{\sqrt{2}+1}\Rightarrow x=1-\frac{1}{\sqrt{2}+1}=\frac{\sqrt{2}}{\sqrt{2}+1}\)

Thay vào hệ phương trình ban đầu,ta có: \(\hept{\begin{cases}\frac{\sqrt{2}}{\sqrt{2}+1}+\frac{\sqrt{2}}{\sqrt{2}+1}.a=3\left(3\right)\\\frac{\sqrt{2}}{\sqrt{2}+1}.a-\frac{\sqrt{1}}{\sqrt{2}+1}=2\left(4\right)\end{cases}}\)

Lấy (3) + (4),vế với vế,ta có: \(\frac{2\sqrt{2}}{\sqrt{2}+1}.a=5\Leftrightarrow a=\frac{10+5\sqrt{2}}{4}\)