Упр.500 ГДЗ Колягин Ткачёва 7 класс (Алгебра)

1) a(b-c)+d(b-c)-7(c-b);
2) x(a-2)+y(2-a)+(2-a);
3) x(x-y)+y(y-x)-3(x-y);
4) a(b-3)+(3-b)-b(3-b).
Выполнить действия:
1) (a^2+2a+1)/(b^2-4)•(b+2)/(a+1)-a/(b+2);
2) (a^2-2a+1)/(b-2) :(a^2-1)/(b^2-4)-(2a-b)/(a+1);
3) (m-1)/(m+1)-m(1-m^2 )/n•n/(m+1)^2 ;
4) (2n+4)/(2-n)-(mn+n^2)/(4-4n+n^2 ) :(m+n)/(4-n^2 ).

1. $$a(b-c)+d(b-c)-7(c-b)=a(b-c)+d(b-c)+7(b-c)$$
   $$=(b-c)(a+d+7).$$

2. $$x(a-2)+y(2-a)+(2-a)=x(a-2)-y(a-2)-(a-2)$$
   $$=(a-2)(x-y-1).$$

3. $$x(x-y)+y(y-x)-3(x-y)=x(x-y)-y(x-y)-3(x-y)$$
   $$=(x-y)(x-y-3).$$

4. $$a(b-3)+(3-b)-b(3-b)=a(b-3)-(b-3)+b(b-3)$$
   $$=(b-3)(a-1+b).$$

1. $$\frac{a^2+2a+1}{b^2-4}\cdot\frac{b+2}{a+1}-\frac{a}{b+2}$$
   $$=\frac{(a+1)^2}{(b-2)(b+2)}\cdot\frac{b+2}{a+1}-\frac{a}{b+2}$$
   $$=\frac{a+1}{b-2}-\frac{a}{b+2}$$
   $$=\frac{(a+1)(b+2)-a(b-2)}{(b-2)(b+2)}$$
   $$=\frac{ab+2a+b+2-ab+2a}{b^2-4}$$
   $$=\frac{4a+b+2}{b^2-4}.$$

2. $$\frac{a^2-2a+1}{b-2}:\frac{a^2-1}{b^2-4}-\frac{2a-b}{a+1}$$
   $$=\frac{(a-1)^2}{b-2}:\frac{(a-1)(a+1)}{(b-2)(b+2)}-\frac{2a-b}{a+1}$$
   $$=\frac{(a-1)^2}{b-2}\cdot\frac{(b-2)(b+2)}{(a-1)(a+1)}-\frac{2a-b}{a+1}$$
   $$=\frac{(a-1)(b+2)}{a+1}-\frac{2a-b}{a+1}$$
   $$=\frac{(a-1)(b+2)-(2a-b)}{a+1}$$
   $$=\frac{ab+2a-b-2-2a+b}{a+1}$$
   $$=\frac{ab-2}{a+1}.$$

3. $$\frac{m-1}{m+1}-\frac{m(1-m^2)}{n}\cdot\frac{n}{(m+1)^2}$$
   $$=\frac{m-1}{m+1}-\frac{m(1-m^2)}{(m+1)^2}$$
   $$=\frac{m-1}{m+1}-\frac{m(1-m)(1+m)}{(m+1)^2}$$
   $$=\frac{m-1}{m+1}-\frac{m(1-m)}{m+1}$$
   $$=\frac{m-1-m(1-m)}{m+1}$$
   $$=\frac{m-1-m+m^2}{m+1}$$
   $$=\frac{m^2-1}{m+1}$$
   $$=\frac{(m-1)(m+1)}{m+1}=m-1.$$

4. $$\frac{2n+4}{2-n}-\frac{mn+n^2}{4-4n+n^2}:\frac{m+n}{4-n^2}$$
   $$=\frac{2n+4}{2-n}-\frac{n(m+n)}{(2-n)^2}:\frac{m+n}{(2-n)(2+n)}$$
   $$=\frac{2n+4}{2-n}-\frac{n(m+n)}{(2-n)^2}\cdot\frac{(2-n)(2+n)}{m+n}$$
   $$=\frac{2n+4}{2-n}-\frac{n(2+n)}{2-n}$$
   $$=\frac{2n+4-n(2+n)}{2-n}$$
   $$=\frac{4-n^2}{2-n}$$
   $$=\frac{(2-n)(2+n)}{2-n}=2+n.$$

Ответ

1) $$\left(b-c\right)\left(a+d+7\right);$$
2) $$\left(a-2\right)\left(x-y-1\right);$$
3) $$\left(x-y\right)\left(x-y-3\right);$$
4) $$\left(b-3\right)\left(a+b-1\right);$$
1) $$\frac{4a+b+2}{b^2-4};$$
2) $$\frac{ab-2}{a+1};$$
3) $$m-1;$$
4) $$n+2.$$