Sau 1 năm số tiền nhận được là:

\(100\cdot\left(1+7,2\%\right)=107,2\)(triệu đồng)

Sau 2 năm số tiền nhận được là:

\(107,2\cdot\left(1+7,2\%\right)=114,9184\)(triệu đồng)

a: Xét ΔAHB vuông tại H có HM là đường cao

nên \(AM\cdot AB=AH^2\left(1\right)\) và \(MA\cdot MB=HM^2\)

Xét ΔAHC vuông tại H có HN là đường cao

nên \(AN\cdot AC=AH^2\left(2\right);NA\cdot NC=NH^2\)

Xét ΔABC vuông tại A có AH là đường cao

nên \(AH^2=HB\cdot HC\left(3\right);AB^2=BH\cdot BC;AC^2=CH\cdot BC\)

Từ (1) và (3) suy ra \(AM\cdot AB=HB\cdot HC\)

b: Từ (1) và (2) suy ra \(AM\cdot AB=AN\cdot AC\)

c: Xét tứ giác AMHN có \(\widehat{AMH}=\widehat{ANH}=\widehat{MAN}=90^0\)

nên AMHN là hình chữ nhật

=>\(HA^2=HM^2+HN^2\)

=>\(HB\cdot HC=MA\cdot MB+NA\cdot NC\)

d: \(\dfrac{AB^2}{AC^2}=\dfrac{BH\cdot BC}{CH\cdot BC}=\dfrac{BH}{CH}\)

=>\(\dfrac{HB}{HC}=\left(\dfrac{AB}{AC}\right)^2\)

e: Xét ΔAHB vuông tại H có HM là đường cao

nên \(BM\cdot BA=BH^2\)

=>\(BM=\dfrac{BH^2}{BA}\)

Xét ΔAHC vuông tại H có HN là đường cao

nên \(CN\cdot CA=CH^2\)

=>\(CN=\dfrac{CH^2}{CA}\)

Xét ΔABC vuông tại A có AH là đường cao

nên \(AH\cdot BC=AB\cdot AC\)

=>\(BC=\dfrac{AB\cdot AC}{AH}\)

\(BM\cdot CN\cdot BC=\dfrac{BH^2}{BA}\cdot\dfrac{CH^2}{CA}\cdot\dfrac{AB\cdot AC}{AH}\)

\(=\dfrac{BH^2}{AH}\cdot CH^2=\dfrac{\left(BH\cdot CH\right)^2}{AH}=\dfrac{AH^4}{AH}=AH^3\)

mà AH=MN(AMHN là hình chữ nhật)

nên \(BM\cdot CN\cdot BC=MN^3\)

a: \(\Delta=\left[2\left(m+1\right)\right]^2-4\cdot2\cdot m\)

\(=\left(2m+2\right)^2-8m=4m^2+8m+4-8m=4m^2+4>0\forall m\)

=>Phương trình luôn có hai nghiệm phân biệt

b: Theo Vi-et, ta có: \(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{b}{a}=\dfrac{-2\left(m+1\right)}{2}=-\left(m+1\right)\\x_1x_2=\dfrac{c}{a}=\dfrac{m}{2}\end{matrix}\right.\)

\(\left\{{}\begin{matrix}x_1< 1\\x_2< 1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x_1+x_2< 2\\x_1\cdot x_2< 1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}-\left(m+1\right)< 2\\\dfrac{m}{2}< 1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m+1>-2\\m< 2\end{matrix}\right.\)

=>-3<m<2

a) 

\(\Delta=\left[2\left(m+1\right)\right]^2-4\cdot2\cdot m=4\left(m^2+2m+1\right)-8m\\ =4m^2+8m+4-8m=4m^2+4\ge4>0\forall x\)

=> Pt luôn có nghiệm với mọi m 

b) Ta có: \(\left\{{}\begin{matrix}x_1< 1\\x_2< 1\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x_1+x_2< 2\\x_1x_2< 1\end{matrix}\right.\) 

Theo vi-ét: \(\left\{{}\begin{matrix}x_1x_2=\dfrac{m}{2}\\x_1+x_2=\dfrac{-2\left(m+1\right)}{2}=-\left(m+1\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}\dfrac{m}{2}< 1\\-\left(m+1\right)< 2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m< 2\\m>-3\end{matrix}\right.\Rightarrow-3< m< 2\)

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

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

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

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

b: \(\left\{{}\begin{matrix}\left(x+1\right)\left(y-1\right)=xy-1\\\left(x-3\right)\left(y+3\right)=xy-3\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}xy-x+y-1=xy-1\\xy+3x-3y-9=xy-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}-x+y=0\\3x-3y=-3+9=6\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=y\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}0=2\left(vôlý\right)\\x=y\end{matrix}\right.\)

vậy: Hệ vô nghiệm

a) 

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

b) 

\(\left\{{}\begin{matrix}\left(x+1\right)\left(y-1\right)=xy-1\\\left(x-3\right)\left(y+3\right)=xy-3\end{matrix}\right. \Leftrightarrow\left\{{}\begin{matrix}xy-x+y-1=xy-1\\xy+3x-3y-9=xy-3\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}-x+y=0\\3x-3y=6\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}-x+y=0\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\x-y=2\end{matrix}\right.\)

mà: 2 khác 0

=> Hpt vô nghiệm 

a: ĐKXĐ: \(x\ne0;y\ne0\)

Đặt \(\dfrac{1}{x}=a;\dfrac{1}{y}=b\)

Hệ phương trình sẽ trở thành: \(\left\{{}\begin{matrix}a-2b=-1\\2a+b=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a-2b=-1\\4a+2b=6\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}a-2b+4a+2b=-1+6\\2a+b=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5a=5\\b=3-2a\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}a=1\\b=3-2=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{x}=1\\\dfrac{1}{y}=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=1\end{matrix}\right.\left(nhận\right)\)

b: ĐKXĐ: \(\left\{{}\begin{matrix}x\ne y\\x\ne-\dfrac{y}{2}\end{matrix}\right.\)

Đặt \(\dfrac{1}{x-y}=a;\dfrac{1}{2x+y}=b\)

Hệ phương trình sẽ trở thành:

\(\left\{{}\begin{matrix}a+b=2\\3a-2b=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2a+2b=4\\3a-2b=-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2a+2b+3a-2b=4-2\\a+b=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5a=2\\b=2-a\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}a=\dfrac{2}{5}\\b=2-\dfrac{2}{5}=\dfrac{10}{5}-\dfrac{2}{5}=\dfrac{8}{5}\end{matrix}\right.\)

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

=>\(\left\{{}\begin{matrix}x-y+2x+y=\dfrac{5}{2}+\dfrac{5}{8}\\x-y=\dfrac{5}{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x=\dfrac{25}{8}\\y=x-\dfrac{5}{2}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{25}{8}:3=\dfrac{25}{24}\\y=\dfrac{25}{24}-\dfrac{5}{2}=\dfrac{25}{24}-\dfrac{60}{24}=-\dfrac{35}{24}\end{matrix}\right.\left(nhận\right)\)

c: ĐKXĐ: \(x\ne1;y\ne-3\)

Đặt \(\dfrac{x}{x-1}=a;\dfrac{1}{y+3}=b\)

Hệ phương trình sẽ trở thành:

\(\left\{{}\begin{matrix}3a-2b=3\\4a+b=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3a-2b=3\\8a+2b=10\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3a-2b+8a+2b=13\\4a+b=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}11a=13\\b=5-4a\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{13}{11}\\b=5-4\cdot\dfrac{13}{11}=\dfrac{55}{11}-\dfrac{52}{11}=\dfrac{3}{11}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}\dfrac{x}{x-1}=\dfrac{13}{11}\\\dfrac{1}{y+3}=\dfrac{3}{11}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}13\left(x-1\right)=11x\\y+3=\dfrac{11}{3}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}13x-13=11x\\y=\dfrac{11}{3}-3=\dfrac{2}{3}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{13}{2}\\y=\dfrac{2}{3}\end{matrix}\right.\left(nhận\right)\)

Bài 3:

a: \(\left\{{}\begin{matrix}\dfrac{x+y}{2}=\dfrac{x-y}{4}\\\dfrac{x}{3}=\dfrac{y}{5}+1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2\left(x+y\right)=x-y\\\dfrac{5x}{15}=\dfrac{3y}{15}+\dfrac{15}{15}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2x+2y=x-y\\5x=3y+15\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x+3y=0\\5x-3y=15\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x+3y+5x-3y=0+15\\x=-3y\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}6x=15\\x=-3y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{15}{6}=\dfrac{5}{2}\\y=\dfrac{x}{-3}=\dfrac{5}{2}:\left(-3\right)=-\dfrac{5}{6}\end{matrix}\right.\)

b: \(\left\{{}\begin{matrix}\left(x-1\right)\left(y+3\right)=xy+27\\\left(x-2\right)\left(y+1\right)=xy+8\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}xy+3x-y-3=xy+27\\xy+x-2y-2=xy+8\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3x-y=30\\x-2y=10\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}6x-2y=60\\x-2y=10\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}6x-2y-x+2y=60-10\\x-2y=10\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}5x=50\\2y=x-10\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=10\\2y=10-10=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=10\\y=0\end{matrix}\right.\)

Giúp mấy bài còn lại giùm mình với ạ

a: \(\left\{{}\begin{matrix}x-3y=4\\2x+3y=17\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-3y+2x+3y=4+17\\x-3y=4\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3x=21\\3y=x-4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=7\\3y=7-4=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=7\\y=1\end{matrix}\right.\)

b: \(\left\{{}\begin{matrix}2x+5y=8\\2x-7y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x+5y-2x+7y=8-0\\2x=7y\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}12y=8\\x=\dfrac{7}{2}y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{8}{12}=\dfrac{2}{3}\\x=\dfrac{7}{2}\cdot\dfrac{2}{3}=\dfrac{7}{3}\end{matrix}\right.\)

c: \(\left\{{}\begin{matrix}3x+5y=14\\-4x+3y=20\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}12x+20y=56\\-12x+9y=60\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}12x+20y-12x+9y=56+60\\3x+5y=14\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}29y=116\\3x=14-5y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=4\\3x=14-5\cdot4=14-20=-6\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=4\\x=-2\end{matrix}\right.\)