154 Вправа 154Алгебра 8 клас Бевз 2025 · § 4 · стор. 36

154. Доведи тотожність.

а) ${\displaystyle \frac{a}{a+c}+\frac{c}{a+c}=}$${\displaystyle \frac{\cancel{\left(a+c\right)}}{\cancel{\left(a+c\right)}}=}$${\displaystyle 1}$;

${\displaystyle 1=}$${\displaystyle 1}$ — тотожність доведено;

б) ${\displaystyle \frac{2a}{a-c}-\frac{c+a}{a-c}=}$${\displaystyle \frac{2a-(c+a)}{a-c}=}$${\displaystyle \frac{2a-a-c}{a-c}=}$${\displaystyle \frac{\cancel{\left(a-c\right)}}{\cancel{\left(a-c\right)}}=}$${\displaystyle 1}$;

${\displaystyle 1=}$${\displaystyle 1}$ — тотожність доведено;

в) ${\displaystyle \frac{x(1+y)}{x-y}-\frac{y(1+x)}{x-y}=}$${\displaystyle \frac{x(1+y)-y(1+x)}{x-y}}$=

=${\displaystyle \frac{xy+x-xy-y}{x-y}=}$${\displaystyle \frac{\cancel{\left(x-y\right)}}{\cancel{\left(x-y\right)}}=}$${\displaystyle 1}$;

${\displaystyle 1=}$${\displaystyle 1}$ — тотожність доведено;

г) ${\displaystyle \frac{a(b-1)}{a-b}-\frac{b(a-1)}{a-b}=}$${\displaystyle \frac{a(b-1)-b(a-1)}{a-b}}$=

=${\displaystyle \frac{ab-a-ab+b}{a-b}=}$${\displaystyle \frac{b-a}{a-b}=}$${\displaystyle \frac{-\cancel{\left(a-b\right)}}{\cancel{\left(a-b\right)}}=}$${\displaystyle -1}$;

${\displaystyle -1=}$${\displaystyle -1}$ — тотожність доведено.