Упражнение 72

\(1)\ \frac{a+b}{c-7}+\frac{a}{7-c}=\frac{a+b}{c-7}+\frac{a}{ {-}(c-7)}=\frac{a+b}{c-7}-\frac{a}{c-7}=\frac{a+b-a}{c-7}=\frac{b}{c-7};\)

\(2)\ \frac{5m}{m-n}+\frac{5n}{n-m}=\frac{5m}{m-n}+\frac{5n}{ {-}(m-n)}=\frac{5m}{m-n}-\frac{5n}{m-n}=\frac{5m-5n}{m-n}=\frac{ 5(m-n)}{m-n}=5;\)

\(3)\ \frac{2x-4y}{x-3y}-\frac{4x-14y}{3y-x}=\frac{2x-4y}{x-3y}-\frac{4x-14y}{ {-}(x-3y)}=\frac{2x-4y}{x-3y}+\frac{4x-14y}{x-3y}=\frac{2x-4y+4x-14y}{x-3y}=\frac{6x-18y}{x-3y}=\frac{ 6(x-3y)}{x-3y}=6;\)

\(4)\ \frac{81b^2}{9b-a}+\frac{a^2}{a-9b}=\frac{81b^2}{9b-a}+\frac{a^2}{ {-}(9b-a)}=\frac{81b^2}{9b-a}-\frac{a^2}{9b-a}=\frac{81b^2-a^2}{9b-a}=\frac{ (9b-a)(9b+a)}{9b-a}=9b+a;\)

\(5)\ \frac{t^2}{3t-6}+\frac{4}{6-3t}=\frac{t^2}{3t-6}+\frac{4}{ {-}(3t-6)}=\frac{t^2}{3t-6}-\frac{4}{3t-6}=\frac{t^2-4}{3t-6}=\frac{ (t-2)(t+2)}{ 3(t-2)}=\frac{t+2}{3};\)

\(6)\ \frac{y^2}{y-1}-\frac{1-2y}{1-y}=\frac{y^2}{y-1}-\frac{1-2y}{ {-}(y-1)}=\frac{y^2}{y-1}+\frac{1-2y}{y-1}=\frac{y^2+1-2y}{y-1}=\frac{y^2-2y+1}{y-1}=\frac{ (y-1)^2}{y-1}=y-1.\)