414 Вправа 414Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 113

414. Знайдіть m/n, якщо:

1) ${\displaystyle \frac{n^{-1}+m^{-1}}{n^{-1}-m^{-1}}=}$${\displaystyle 3}$; множимо чисельник і знаменник на ${\displaystyle mn}$:

${\displaystyle \frac{m+n}{m-n}=}$${\displaystyle 3}$ | ${\displaystyle \cdot \left(m-n\right)}$

${\displaystyle \frac{(m+n)\cdot \cancel{\left(m-n\right)}}{\cancel{\left(m-n\right)}}=}$${\displaystyle 3(m-n)}$;

${\displaystyle m+n=}$${\displaystyle 3m-3n}$;

${\displaystyle n+3n=}$${\displaystyle 3m-m}$;

${\displaystyle 4n=}$${\displaystyle 2m}$ | ${\displaystyle :2n}$

${\displaystyle \frac{4\cancel{n}}{2\cancel{n}}=}$${\displaystyle \frac{\cancel{2}m}{\cancel{2}n}}$;

${\displaystyle \frac{m}{n}=}$${\displaystyle \frac{4}{2}=}$${\displaystyle 2}$;

2) ${\displaystyle \frac{m^{-1}-4n^{-1}}{3n^{-1}+m^{-1}}=}$${\displaystyle 5}$; множимо чисельник і знаменник на ${\displaystyle mn}$:

${\displaystyle \frac{n-4m}{3m+n}=}$${\displaystyle 5}$ | ${\displaystyle \cdot \left(3m+n\right)}$

${\displaystyle \frac{(n-4m)\cdot \cancel{\left(3m+n\right)}}{\cancel{\left(3m+n\right)}}=}$${\displaystyle 5(3m+n)}$;

${\displaystyle n-4m=}$${\displaystyle 15m+5n}$;

${\displaystyle -4m-15m=}$${\displaystyle 5n-n}$;

${\displaystyle -19m=}$${\displaystyle 4n}$ | ${\displaystyle :\left(-19n\right)}$

${\displaystyle \frac{\cancel{-19}m}{\cancel{-19}n}=}$${\displaystyle \frac{4\cancel{n}}{-19\cancel{n}}}$;

${\displaystyle \frac{m}{n}=}$${\displaystyle -\frac{4}{19}}$.