№ 7.2 Алгебра = № 12.2 Математика
Виконайте дії:
1. $\frac{10x\ +\ y}{5x}-\frac{{3y}^2}{x^2}\cdot\frac{x}{15y};$
2. $\frac{a^2-4}{9-b^2}:\frac{a\ -\ 2}{3\ +\ b}-\frac{2}{3\ -\ b};$
3. $\frac{x\ +\ y}{3x-y}+\frac{1}{x+y}\cdot\frac{x^2-y^2}{3x-y};$
4. $\ m+\frac{m^2+mn}{n-m}\cdot\frac{m}{m+n}.$
Розв'язок:
1. $\frac{10x\ +\ y}{5x}-\frac{{3y}^2}{x^2}\cdot\frac{x}{15y}=$
$= \frac{10x\ +\ y}{5x}-\frac{{3y}^2\cdot x}{x^2\cdot15y}=$
$= \frac{10x\ +\ y}{5x}-\frac{y}{5x}=\frac{10x\ +\ y-y}{5x}=$
$= \frac{10x}{5x}=2;$
2. $\frac{a^2-4}{9-b^2}:\frac{a\ -\ 2}{3\ +\ b}-\frac{2}{3\ -\ b}=$
$= \frac{\left(a-2\right)\left(a+2\right)\left(3+b\right)}{\left(3-b\right)\left(3+b\right)\left(a-2\right)}-\frac{2}{3\ -\ b}=$
$= \frac{a\ +\ 2}{3\ -\ b}-\frac{2}{3\ -\ b}=\frac{a\ +\ 2-2}{3\ -\ b}=$
$= \frac{a}{3\ -\ b};$
3. $\frac{x\ +\ y}{3x-y}+\frac{1}{x+y}\cdot\frac{x^2-y^2}{3x-y}=$
$= \frac{x\ +\ y}{3x-y}+\frac{1\cdot\left(x-y\right)\left(x+y\right)}{\left(x+y\right)\left(3x-y\right)}=$
$= \frac{x\ +\ y}{3x-y}+\frac{x\ -\ y}{3x-y}=\frac{x\ +\ y+x-y}{3x-y}=$
$= \frac{2x}{3x-y};$
4. $\ m+\frac{m^2+mn}{n-m}\cdot\frac{m}{m+n}=$
$= m+\frac{m\left(m+n\right)m}{\left(n-m\right)\left(m+n\right)}=$
$= m+\frac{m^2}{n-m}=\frac{mn-m^2+m^2}{n-m}=$
$= \frac{mn}{n-m}.$