1:
\(\dfrac{a-b}{a+b}-\dfrac{a+b}{a-b}\)
\(=\dfrac{\left(a-b\right)^2-\left(a+b\right)^2}{\left(a+b\right)\left(a-b\right)}\)
\(=\dfrac{a^2-2ab+b^2-a^2-2ab-b^2}{a^2-b^2}=\dfrac{-4ab}{a^2-b^2}\)
2:
\(\dfrac{x+y}{xy}-\dfrac{y+z}{yz}\)
\(=\dfrac{z\left(x+y\right)-x\left(y+z\right)}{xyz}\)
\(=\dfrac{xz+yz-xy-xz}{xyz}=\dfrac{yz-xz}{xyz}\)
\(=\dfrac{z\left(y-x\right)}{xyz}=\dfrac{y-x}{xy}\)
3: \(\dfrac{1}{x-2}+\dfrac{2}{x^2-4x+4}\)
\(=\dfrac{1}{x-2}+\dfrac{2}{\left(x-2\right)^2}\)
\(=\dfrac{x-2+2}{\left(x-2\right)^2}=\dfrac{x}{\left(x-2\right)^2}\)
4: \(\dfrac{1}{x+5}+\dfrac{1}{5-x}+\dfrac{2x}{x^2-25}\)
\(=\dfrac{1}{x+5}-\dfrac{1}{x-5}+\dfrac{2x}{\left(x+5\right)\left(x-5\right)}\)
\(=\dfrac{x-5-x-5+2x}{\left(x-5\right)\left(x+5\right)}\)
\(=\dfrac{2x-10}{\left(x-5\right)\left(x+5\right)}=\dfrac{2}{x+5}\)
5: \(x+\dfrac{2y^2}{x+y}-y\)
\(=\dfrac{\left(x-y\right)\left(x+y\right)+2y^2}{x+y}\)
\(=\dfrac{x^2-y^2+2y^2}{x+y}=\dfrac{x^2+y^2}{x+y}\)
6:
\(\dfrac{xy+5}{x^2+xy}-\dfrac{y}{x+y}\)
\(=\dfrac{xy+5}{x\left(x+y\right)}-\dfrac{y}{x+y}\)
\(=\dfrac{xy+5-xy}{x\left(x+y\right)}=\dfrac{5}{x\left(x+y\right)}\)
7: \(\dfrac{3y-2x}{x-2y}-\dfrac{x-y}{2y-x}\)
\(=\dfrac{3y-2x}{x-2y}+\dfrac{x-y}{x-2y}\)
\(=\dfrac{-2x+3y+x-y}{x-2y}=\dfrac{-x+2y}{x-2y}=-1\)
8: \(\dfrac{4x+2}{4x-4}+\dfrac{3-6x}{6x-6}\)
\(=\dfrac{2\left(2x+1\right)}{4\left(x-1\right)}+\dfrac{3\left(1-2x\right)}{6\left(x-1\right)}\)
\(=\dfrac{2x+1}{2\left(x-1\right)}+\dfrac{1-2x}{2\left(x-1\right)}\)
\(=\dfrac{2x+1+1-2x}{2\left(x-1\right)}=\dfrac{2}{2\left(x-1\right)}=\dfrac{1}{x-1}\)
9: \(\dfrac{y}{2x^2-xy}+\dfrac{4x}{y^2-2xy}\)
\(=\dfrac{y}{x\left(2x-y\right)}+\dfrac{4x}{y\left(y-2x\right)}\)
\(=\dfrac{y}{x\left(2x-y\right)}-\dfrac{4x}{y\left(2x-y\right)}\)
\(=\dfrac{y^2-4x^2}{xy\left(2x-y\right)}\)
\(=\dfrac{-\left(4x^2-y^2\right)}{xy\left(2x-y\right)}\)
\(=\dfrac{-\left(2x-y\right)\left(2x+y\right)}{xy\left(2x-y\right)}=\dfrac{-2x-y}{xy}\)
10:
\(\dfrac{x}{x-y}+\dfrac{y}{x+y}-\dfrac{2y^2}{y^2-x^2}\)
\(=\dfrac{x\left(x+y\right)+y\left(x-y\right)+2y^2}{\left(x-y\right)\left(x+y\right)}\)
\(=\dfrac{x^2+xy+xy-y^2+2y^2}{\left(x-y\right)\left(x+y\right)}\)
\(=\dfrac{x^2+2xy+y^2}{\left(x-y\right)\left(x+y\right)}=\dfrac{\left(x+y\right)^2}{\left(x-y\right)\left(x+y\right)}=\dfrac{x+y}{x-y}\)
