a) \(Đkxđ:x>0;y>0;x\ne y.\)

b)

 \(A=\left(\sqrt{x}+\dfrac{y-\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\right):\dfrac{x\sqrt{xy}+y\sqrt{xy}}{\sqrt{xy}\left(y-x\right)}.\\ A=\dfrac{x-\sqrt{xy}+y-\sqrt{xy}}{\sqrt{x}-\sqrt{y}}:\dfrac{\sqrt{xy}\left(x+y\right)}{\sqrt{xy}\left(y-x\right)}.\\ A=\dfrac{x+y}{\sqrt{x}-\sqrt{y}}.\dfrac{y-x}{x+y}.\\ A=\dfrac{y-x}{\sqrt{x}-\sqrt{y}}.\\ A=\dfrac{-\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}.\\ A=-\sqrt{x}-\sqrt{y}.\)

\(A=\left(71+x\right)+\left(-24-x\right)+\left(35-x\right).\\A =71+x-24-x+35-x.\\ =82-x.\\ B=x-31-\left[\left(15+x\right)-\left(23-x\right)\right].\\ B=x-31-15-x+23-x.\\ B=-x-23.\\ C=\left(-15+x\right)+\left[25-\left(-x\right)\right].\\ C=-15+x+25+x.\\ C=10+2x.\)

Xét \(\Delta ABC\) vuông tại A:

Trung tuyến AD \(\text{(H,D ϵ BC).}\)

\(\Rightarrow BD=DC=AD=\dfrac{BC}{2}\) (T/c đường trung tuyến trong tam giác vuông).

Xét \(\Delta ADB:AD=BD\left(cmt\right).\)

\(\Rightarrow\Delta ADB\) cân tại D.

\(\Rightarrow\widehat{BAD}=\widehat{ABD}\) (T/c tam giác cân).

Mà \(\left\{{}\begin{matrix}\widehat{F}+\widehat{BAD}=90^o.\\\widehat{C}+\widehat{ABD}=90^o.\end{matrix}\right.\)

\(\Rightarrow\widehat{C}=\widehat{F}.\)

Xét \(\Delta DCE\) và \(\Delta DFB:\)

\(\widehat{C}=\widehat{F}\left(cmt\right).\)

\(\widehat{EDC}=\widehat{BDF}\) (Đối đỉnh).

\(\Rightarrow\Delta DCE\) \(\sim\Delta DFB\left(g-g\right).\)

\(1.\\ A=23\times\left(-24\right)+24\times\left(-77\right).\\ =-24\left(23+77\right).\\ =-24.100=-2400.\\ B=-31\times48-38\times24.\\ =-31\times48-19\times48.\\ =48\left(-31-19\right).\\ 48.\left(-50\right)=-2400.\\ C=1\times3\times5^3\times\left(-2\right)\times\left(-8\right).\\ =3\times125\times16.\\ =48\times125=6000.\)

\(2.\\ a,2x-5=-17.\\ \Leftrightarrow x=\dfrac{-17+5}{2}=-6.\\ b,5-8x=-27.\\ \Leftrightarrow-8x=32.\\ \Leftrightarrow x=-4.\\ c,2\left(x-1\right)+3\left(x+4\right)=50.\\ \Leftrightarrow2x-2+3x+12-50=0.\\ \Leftrightarrow5x=40.\\ \Leftrightarrow x=8.\\ d,18-4:x=14.\\ \Leftrightarrow4:x=4.\\ \Leftrightarrow x=1.\)

\(a,\left(\dfrac{-7}{15}-\dfrac{27}{70}\right)-\left(\dfrac{8}{15}+\dfrac{43}{70}\right).\\ =\dfrac{-7}{15}-\dfrac{27}{70}-\dfrac{8}{15}-\dfrac{43}{70}.\\ =\left(\dfrac{-7}{15}-\dfrac{8}{15}\right)-\left(\dfrac{27}{70}+\dfrac{43}{70}\right).\\ =-1-1=-2.\)

\(b,\dfrac{3}{7}+\left(\dfrac{-1}{5}+\dfrac{-3}{7}\right).\\ =\dfrac{3}{7}+\dfrac{-1}{5}+\dfrac{-3}{7}.\\ =0+\dfrac{-1}{5}=-\dfrac{1}{5}.\\ c,\left(4-\dfrac{12}{5}\right).\dfrac{25}{8}-\dfrac{2}{5}:\dfrac{-4}{25}.\\ =\dfrac{25}{2}-\dfrac{15}{2}+\dfrac{5}{2}.\\ =5+\dfrac{5}{2}=\dfrac{15}{2}.\)

\(\Leftrightarrow\dfrac{1}{4}\left(2x-1\right)=\dfrac{5}{8}.\\ \Leftrightarrow2x-1=\dfrac{5}{2}.\\ \Leftrightarrow x=\dfrac{7}{4}.\)

\(\Rightarrow-3x=-2y.\\ \Leftrightarrow3x=2y.\\ \Leftrightarrow x=\dfrac{2}{3}y.\\ 4x-3y=9.\\ \Rightarrow4.\dfrac{2}{3}y-3y=9.\\ \Leftrightarrow\dfrac{8}{3}y-3y=9.\\ \Leftrightarrow-\dfrac{1}{3}y=9.\\ \Leftrightarrow y=-27.\\ \Rightarrow x=\dfrac{2}{3}y=\dfrac{2}{3}.\left(-27\right)=-18.\)

\(\Rightarrow x=\dfrac{3}{4}y.\\ x-y=11.\\ \Rightarrow\dfrac{3}{4}y-y=11.\\ \Leftrightarrow-\dfrac{1}{4}y=11.\\ \Leftrightarrow y=-44.\\ \Rightarrow x=\dfrac{3}{4}y=\dfrac{3}{4}.\left(-44\right)=-33.\)

\(\Rightarrow x=\dfrac{4}{3}y.\\ 2x+3y=34.\\ \Rightarrow2.\dfrac{4}{3}y+3y=34.\\ \Leftrightarrow\dfrac{8}{3}y+3y=34.\\ \Leftrightarrow\dfrac{17}{3}y=34.\\ \Leftrightarrow y=6.\)

\(\Leftrightarrow1,2.50=\left(3x+4\right).30.\\ \Leftrightarrow60=90x+120.\\ \Leftrightarrow90x=60-120.\\ \Leftrightarrow90x=-60.\\ \Leftrightarrow x=\dfrac{-2}{3}.\)

\(x\ne1.\)

\(\Leftrightarrow3.16=21\left(x-1\right).\\ \Leftrightarrow48=21x-21.\\ \Leftrightarrow21x=69.\\ \Leftrightarrow x=\dfrac{23}{7}.\)

\(\\ \Rightarrow x=\dfrac{5}{2}y.\) 

\(3x-2y=44.\\ \Rightarrow3.\dfrac{5}{2}y-2y=44.\\ \Leftrightarrow\dfrac{15}{2}y-2y=44.\\ \Leftrightarrow\dfrac{11}{2}y=44.\\ \Leftrightarrow y=8.\)

\(\Rightarrow x=\dfrac{5}{2}y=\dfrac{5}{2}.8=20.\)