247° Вправа 247Алгебра 8 клас Тарасенкова 2025 · § 7 · стор. 75

247. Застосуйте основну властивість пропорції та розв’яжіть рівняння:

Якщо ${\displaystyle \frac{A}{B}=}$${\displaystyle \frac{C}{D}}$, то ${\displaystyle A\cdot D=}$${\displaystyle B\cdot C}$.

1) ${\displaystyle \frac{2}{x}=}$${\displaystyle \frac{5}{7}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 5\cdot x=}$${\displaystyle 2\cdot 7}$;

${\displaystyle 5x=}$${\displaystyle 14}$ | ${\displaystyle :5}$

${\displaystyle x=}$${\displaystyle 14:5}$;

${\displaystyle x=}$${\displaystyle 2,8}$;

2) ${\displaystyle \frac{3}{2x}=}$${\displaystyle \frac{-4}{5}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle -4\cdot 2x=}$${\displaystyle 3\cdot 5}$;

${\displaystyle -8x=}$${\displaystyle 15}$ | ${\displaystyle :\left(-8\right)}$

${\displaystyle x=}$${\displaystyle -\frac{15}{8}}$;

${\displaystyle x=}$${\displaystyle -1,875}$;

3) ${\displaystyle \frac{4}{3x}=}$${\displaystyle \frac{-8}{27}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle -8\cdot 3x=}$${\displaystyle 4\cdot 27}$;

${\displaystyle -24x=}$${\displaystyle 108}$ | ${\displaystyle :\left(-24\right)}$

${\displaystyle x=}$${\displaystyle -4,5}$;

4) ${\displaystyle \frac{-5}{16x}=}$${\displaystyle \frac{25}{32}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 25\cdot 16x=}$${\displaystyle -5\cdot 32}$;

${\displaystyle 400x=}$${\displaystyle -160}$ | ${\displaystyle :400}$

${\displaystyle x=}$${\displaystyle -0,4}$;

5) ${\displaystyle \frac{x-1}{x}=}$${\displaystyle \frac{3}{4}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 4\cdot (x-1)=}$${\displaystyle 3\cdot x}$;

${\displaystyle 4x-4=}$${\displaystyle 3x}$;

${\displaystyle 4x-3x=}$${\displaystyle 4}$;

${\displaystyle x=}$${\displaystyle 4}$;

6) ${\displaystyle \frac{x-2}{2x}=}$${\displaystyle \frac{2}{3}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 3\cdot (x-2)=}$${\displaystyle 2\cdot 2x}$;

${\displaystyle 3x-6=}$${\displaystyle 4x}$;

${\displaystyle 3x-4x=}$${\displaystyle 6}$;

${\displaystyle -x=}$${\displaystyle 6}$ | ${\displaystyle \cdot \left(-1\right)}$

${\displaystyle x=}$${\displaystyle -6}$;

7) ${\displaystyle \frac{1+x}{2x}=}$${\displaystyle \frac{5}{6}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 6\cdot (1+x)=}$${\displaystyle 5\cdot 2x}$;

${\displaystyle 6+6x=}$${\displaystyle 10x}$;

${\displaystyle 6x-10x=}$${\displaystyle -6}$;

${\displaystyle -4x=}$${\displaystyle -6}$ | ${\displaystyle :\left(-4\right)}$

${\displaystyle x=}$${\displaystyle 1,5}$;

8) ${\displaystyle \frac{4-x}{5x}=}$${\displaystyle \frac{9}{10}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 10\cdot (4-x)=}$${\displaystyle 9\cdot 5x}$;

${\displaystyle 40-10x=}$${\displaystyle 45x}$;

${\displaystyle -10x-45x=}$${\displaystyle -40}$;

${\displaystyle -55x=}$${\displaystyle -40}$ | ${\displaystyle :\left(-55\right)}$

${\displaystyle x=}$${\displaystyle \frac{\stackrel{8}{\cancel{40}}}{\underset{11}{\cancel{55}}}}$;

${\displaystyle x=}$${\displaystyle \frac{8}{11}}$;

9) ${\displaystyle \frac{3+x}{3x}=}$${\displaystyle \frac{4}{7}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 7\cdot (3+x)=}$${\displaystyle 4\cdot 3x}$;

${\displaystyle 21+7x=}$${\displaystyle 12x}$;

${\displaystyle 7x-12x=}$${\displaystyle -21}$;

${\displaystyle -5x=}$${\displaystyle -21}$ | ${\displaystyle :\left(-5\right)}$

${\displaystyle x=}$${\displaystyle 4,2}$;

10) ${\displaystyle \frac{3-x}{3x}=}$${\displaystyle \frac{8}{9}}$;

ОДЗ: ${\displaystyle x\ne 0}$;

${\displaystyle 9\cdot (3-x)=}$${\displaystyle 8\cdot 3x}$;

${\displaystyle 27-9x=}$${\displaystyle 24x}$;

${\displaystyle -9x-24x=}$${\displaystyle -27}$;

${\displaystyle -33x=}$${\displaystyle -27}$ | ${\displaystyle :\left(-33\right)}$

${\displaystyle x=}$${\displaystyle \frac{\stackrel{9}{\cancel{27}}}{\underset{11}{\cancel{33}}}}$;

${\displaystyle x=}$${\displaystyle \frac{9}{11}}$;

11) ${\displaystyle \frac{x}{3+x}=}$${\displaystyle \frac{3}{5}}$;

ОДЗ: ${\displaystyle 3+x\ne 0}$;

${\displaystyle x\ne -3}$;

${\displaystyle 5\cdot x=}$${\displaystyle 3\cdot (3+x)}$;

${\displaystyle 5x=}$${\displaystyle 9+3x}$;

${\displaystyle 5x-3x=}$${\displaystyle 9}$;

${\displaystyle 2x=}$${\displaystyle 9}$ | ${\displaystyle :2}$

${\displaystyle x=}$${\displaystyle 4,5}$;

12) ${\displaystyle \frac{-x}{2+x}=}$${\displaystyle \frac{6}{11}}$;

ОДЗ: ${\displaystyle 2+x\ne 0}$;

${\displaystyle x\ne -2}$;

${\displaystyle 11\cdot (-x)=}$${\displaystyle 6\cdot (2+x)}$;

${\displaystyle -11x=}$${\displaystyle 12+6x}$;

${\displaystyle -11x-6x=}$${\displaystyle 12}$;

${\displaystyle -17x=}$${\displaystyle 12}$ | ${\displaystyle :\left(-17\right)}$

${\displaystyle x=}$${\displaystyle -\frac{12}{17}}$;

13) ${\displaystyle \frac{x}{5+x}=}$${\displaystyle \frac{2}{7}}$;

ОДЗ: ${\displaystyle 5+x\ne 0}$;

${\displaystyle x\ne -5}$;

${\displaystyle 7\cdot x=}$${\displaystyle 2\cdot (5+x)}$;

${\displaystyle 7x=}$${\displaystyle 10+2x}$;

${\displaystyle 7x-2x=}$${\displaystyle 10}$;

${\displaystyle 5x=}$${\displaystyle 10}$ | ${\displaystyle :5}$

${\displaystyle x=}$${\displaystyle 2}$;

14) ${\displaystyle \frac{4x}{4-x}=}$${\displaystyle \frac{5}{8}}$;

ОДЗ: ${\displaystyle 4-x\ne 0}$;

${\displaystyle x\ne 4}$;

${\displaystyle 8\cdot 4x=}$${\displaystyle 5\cdot (4-x)}$;

${\displaystyle 32x=}$${\displaystyle 20-5x}$;

${\displaystyle 32x+5x=}$${\displaystyle 20}$;

${\displaystyle 37x=}$${\displaystyle 20}$ | ${\displaystyle :37}$

${\displaystyle x=}$${\displaystyle \frac{20}{37}}$;

15) ${\displaystyle \frac{2x}{4-x}=}$${\displaystyle \frac{-2}{7}}$;

ОДЗ: ${\displaystyle 4-x\ne 0}$;

${\displaystyle x\ne 4}$;

${\displaystyle 7\cdot 2x=}$${\displaystyle -2\cdot (4-x)}$;

${\displaystyle 14x=}$${\displaystyle -8+2x}$;

${\displaystyle 14x-2x=}$${\displaystyle -8}$;

${\displaystyle 12x=}$${\displaystyle -8}$ | ${\displaystyle :12}$

${\displaystyle x=}$${\displaystyle -\frac{\stackrel{2}{\cancel{8}}}{\underset{3}{\cancel{12}}}}$;

${\displaystyle x=}$${\displaystyle -\frac{2}{3}}$;

16) ${\displaystyle \frac{-3x}{2-x}=}$${\displaystyle \frac{4}{9}}$;

ОДЗ: ${\displaystyle 2-x\ne 0}$;

${\displaystyle x\ne 2}$;

${\displaystyle 9\cdot (-3x)=}$${\displaystyle 4\cdot (2-x)}$;

${\displaystyle -27x=}$${\displaystyle 8-4x}$;

${\displaystyle -27x+4x=}$${\displaystyle 8}$;

${\displaystyle -23x=}$${\displaystyle 8}$ | ${\displaystyle :\left(-23\right)}$

${\displaystyle x=}$${\displaystyle -\frac{8}{23}}$;

17) ${\displaystyle \frac{x-1}{2+x}=}$${\displaystyle \frac{2}{3}}$;

ОДЗ: ${\displaystyle 2+x\ne 0}$;

${\displaystyle x\ne -2}$;

${\displaystyle 3\cdot (x-1)=}$${\displaystyle 2\cdot (2+x)}$;

${\displaystyle 3x-3=}$${\displaystyle 4+2x}$;

${\displaystyle 3x-2x=}$${\displaystyle 4+3}$;

${\displaystyle x=}$${\displaystyle 7}$;

18) ${\displaystyle \frac{2-x}{3-2x}=}$${\displaystyle \frac{5}{7}}$;

ОДЗ: ${\displaystyle 3-2x\ne 0}$;

${\displaystyle x\ne 1,5}$;

${\displaystyle 7\cdot (2-x)=}$${\displaystyle 5\cdot (3-2x)}$;

${\displaystyle 14-7x=}$${\displaystyle 15-10x}$;

${\displaystyle -7x+10x=}$${\displaystyle 15-14}$;

${\displaystyle 3x=}$${\displaystyle 1}$ | ${\displaystyle :3}$

${\displaystyle x=}$${\displaystyle \frac{1}{3}}$.