413 Вправа 413Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 113

413. Знайдіть x/y, якщо:

1) ${\displaystyle \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}}=}$${\displaystyle 2}$; множимо чисельник і знаменник на ${\displaystyle xy}$:

${\displaystyle \frac{y+x}{y-x}=}$${\displaystyle 2}$ | ${\displaystyle \cdot \left(y-x\right)}$

${\displaystyle \frac{(y+x)\cdot \cancel{\left(y-x\right)}}{\cancel{\left(y-x\right)}}=}$${\displaystyle 2(y-x)}$;

${\displaystyle y+x=}$${\displaystyle 2y-2x}$;

${\displaystyle x+2x=}$${\displaystyle 2y-y}$;

${\displaystyle 3x=}$${\displaystyle y}$ | ${\displaystyle :3y}$

${\displaystyle \frac{\cancel{3}x}{\cancel{3}y}=}$${\displaystyle \frac{\cancel{y}}{3\cancel{y}}}$;

${\displaystyle \frac{x}{y}=}$${\displaystyle \frac{1}{3}}$;

2) ${\displaystyle \frac{2x^{-1}-y^{-1}}{x^{-1}+2y^{-1}}=}$${\displaystyle 5}$; множимо чисельник і знаменник на ${\displaystyle xy}$:

${\displaystyle \frac{2y-x}{y+2x}=}$${\displaystyle 5}$ | ${\displaystyle \cdot \left(y+2x\right)}$

${\displaystyle \frac{(2y-x)\cdot \cancel{\left(y+2x\right)}}{\cancel{\left(y+2x\right)}}=}$${\displaystyle 5(y+2x)}$;

${\displaystyle 2y-x=}$${\displaystyle 5y+10x}$;

${\displaystyle -x-10x=}$${\displaystyle 5y-2y}$;

${\displaystyle -11x=}$${\displaystyle 3y}$ | ${\displaystyle :\left(-11y\right)}$

${\displaystyle \frac{\cancel{-11}x}{\cancel{-11}y}=}$${\displaystyle \frac{3\cancel{y}}{-11\cancel{y}}}$;

${\displaystyle \frac{x}{y}=}$${\displaystyle -\frac{3}{11}}$.