Решебник по алгебре 8 класс Мерзляк ФГОС Задание 843

 

\[\boxed{\mathbf{843\ (843).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ \frac{3x - 6y}{3x} = \frac{3(x - 2y)}{3x} =\]

\[= \frac{x - 2y}{x}\]

\[2)\ \frac{3a + 9b}{4a + 12b} = \frac{3 \cdot (a + 3b)}{4 \cdot (a + 3b)} = \frac{3}{4}\]

\[3)\ \frac{a^{2} - 49}{3a + 21} = \frac{(a - 7)(a + 7)}{3 \cdot (a + 7)} =\]

\[= \frac{a - 7}{3}\]

\[4)\ \frac{12x^{2} - 4x}{2 - 6x} = \frac{- 4x(3x - 1)}{2 \cdot (1 - 3x)} =\]

\[= - 2x\]

\[5)\ \frac{x^{2} - 9}{x^{2} + 6x + 9} =\]

\[= \frac{(x - 3)(x + 3)}{(x + 3)^{2}} = \frac{x - 3}{x + 3}\]

\[6)\ \frac{b^{7} + b^{4}}{b^{2} + b^{5}} = \frac{b^{4}\left( b^{3} + 1 \right)}{b^{2}\left( 1 + b^{3} \right)} = b²\]

\[7)\ \frac{a^{3} + 64}{3a + 12} =\]

\[= \frac{(a + 4)\left( a^{2} - 4a + 16 \right)}{3 \cdot (a + 4)} =\]

\[= \frac{a² - 4a + 16}{3}\]

\[8)\ \frac{xb - 5y + 5b - xy}{x^{2} - 25} =\]

\[= \frac{x(b - y) + 5 \cdot (b - y)}{(x - 5)(x + 5)} =\]

\[= \frac{(b - y)(x + 5)}{(x - 5)(x + 5)} = \frac{b - y}{x - 5}\]

\[9)\ \frac{7m^{2} - 7m + 7}{14m^{3} + 14} =\]

\[= \frac{7 \cdot \left( m^{2} - m + 1 \right)}{14 \cdot (m + 1)\left( m^{2} - m + 1 \right)} =\]

\[= \frac{1}{2m + 2}\]

\[10)\ \frac{a^{2} + bc - b^{2} + ac}{ab + c^{2} + ac - b^{2}} =\]

\[= \frac{(a - b)(a + b) + c(a + b)}{(c - b)(c + b) + a(b + c)} =\]

\[= \frac{(a + b)(a - b + c)}{(b + c)(a - b + c)} = \frac{a + b}{b + c}\]

\[11)\ \frac{20mn^{2} - 20m^{2}n + 5m^{3}}{10mn - 5m^{2}} =\]

\[= \frac{5m\left( 4n^{2} - 4mn + m^{2} \right)}{5m(2n - m)} =\]

\[= \frac{5m(2n - m)^{2}}{5m(2n - m)} = 2n - m\]

\[12)\ \frac{x^{2} - yz + xz - y^{2}}{x^{2} + yz - xz - y^{2}} =\]

\[= \frac{(x - y)(x + y) + z(x - y)}{(x - y)(x + y) - z(x - y)} =\]

\[= \frac{(x - y)(x + y + z)}{(x - y)(x + y - z)} =\]

\[= \frac{x + y + z}{x + y - z}\]

\[\boxed{\mathbf{8}\mathbf{4}\mathbf{3}\mathbf{\text{.\ }}Еуроки\ - \ ДЗ\ без\ мороки}\]

\[Пусть\ x - числитель,\]

\[а\ (x + 5) - знаменатель.\]

\[Тогда\ новая\ дробь\ \frac{x - 3}{x + 9}\text{.\ }\]

\[По\ условию\ известно,\ что\ \]

\[новая\ дробь\ на\ \frac{1}{3}\ меньше.\]

\[Составляем\ уравнение:\]

\[\frac{x}{x + 5} - \frac{x - 3}{x + 9} - \frac{1}{3} = 0\]

\[\frac{- x(x - 7)}{3 \cdot (x + 3)(x + 9)} = 0\]

\[x = 0 - не\ удовлетворяет\ \]

\[условию.\]

\[x = 7,\ тогда\ \frac{7}{12} - исходная\ \]

\[дробь.\]

\[Ответ:\ \frac{7}{12}.\]