номер 118 гдз 8 класс алгебра Муравин Муравин

1)\begin{equation}\left ( 1-\frac{6a-58}{a^{2}-49} \right ):\left ( \frac{2}{a-7}+\frac{5}{a+7} \right ) \end{equation} 1)\begin{equation}1-\frac{6a-58}{a^{2}-49}=\frac{a^{2}-49-6a+58}{a^{2}-49}= \end{equation} \begin{equation}=\frac{a^{2}-6a+9}{a^{2}-49}=\frac{\left ( a-3 \right )^{2}}{a^{2}-49} \end{equation} 2)\begin{equation}\frac{2}{a-7}+\frac{5}{a+7}= \end{equation} \begin{equation}=\frac{2a+14+5a-35}{a^{2}-49}= \end{equation} \begin{equation}=\frac{7a-21}{a^{2}-49}=\frac{7\left ( a-3 \right )}{a^{2}-49} \end{equation} 3)\begin{equation}\frac{\left ( a-3 \right )^{2}}{a^{2}-49}:\frac{7\left ( a-3 \right )}{a^{2}-49}= \end{equation} \begin{equation}=\frac{\left ( a-3 \right )^{2}}{a^{2}-49}\cdot \frac{a^{2}-49}{7\left ( a-3 \right )}= \end{equation} \begin{equation}=\frac{a-3}{7} \end{equation} 2)\begin{equation}\left ( \frac{5}{x-6}+\frac{3}{x+6} \right ):\left ( 4+\frac{153-12x}{x^{2}-36} \right ) \end{equation} 1)\begin{equation}\frac{5}{x-6}+\frac{3}{x+6}= \end{equation} \begin{equation}=\frac{5x+30+3x-18}{x^{2}-36}= \end{equation} \begin{equation}=\frac{8x+18}{x^{2}-36}=\frac{4\left ( 2x+3 \right )}{x^{2}-36} \end{equation} 2)\begin{equation}4+\frac{153-12x}{x^{2}-36}= \end{equation} \begin{equation}=\frac{4x^{2}-144+153-12x}{x^{2}-36}= \end{equation} \begin{equation}=\frac{4x^{2}-12x+9}{x^{2}-36}=\frac{\left ( 2x-3 \right )^{2}}{x^{2}-36} \end{equation} 3)\begin{equation}\frac{4\left ( 2x+3 \right )}{x^{2}-36}:\frac{\left ( 2x-3 \right )^{2}}{x^{2}-36}= \end{equation} \begin{equation}=\frac{4\left ( 2x+3 \right )}{x^{2}-36}\cdot \frac{x^{2}-36}{\left ( 2x-3 \right )^{2}}= \end{equation} \begin{equation}=\frac{4\left ( 2x+3 \right )}{\left ( 2x-3 \right )^{2}} \end{equation} 3)\begin{equation}\frac{1}{2}-\frac{x^{3}-16x}{2x-24}\cdot \left ( \frac{2}{x^{2}-16}-\frac{1}{\left ( x-4 \right )^{2}} \right ) \end{equation} 1)\begin{equation}\frac{2}{x^{2}-16}-\frac{1}{\left ( x-4 \right )^{2}}= \end{equation} \begin{equation}=\frac{2x-8-x-4}{\left ( x-4 \right )^{2}\cdot \left ( x+4 \right )}= \end{equation} \begin{equation}=\frac{x-12}{\left ( x-4 \right )^{2}\cdot \left ( x+4 \right )} \end{equation} 2)\begin{equation}\frac{x^{3}-16x}{2x-24}\cdot \frac{x-12}{\left ( x-4 \right )^{2}\left ( x+4 \right )}= \end{equation} \begin{equation}=\frac{x\left ( x^{2}-16 \right )}{2\left ( x-12 \right )}\cdot \frac{x-12}{\left ( x-4 \right )^{2}\cdot \left ( x+4 \right )}= \end{equation} \begin{equation}=\frac{x\left ( x-4 \right )\left ( x+4 \right )}{2\cdot \left ( x-4 \right )^{2}\cdot \left ( x+4 \right )}= \end{equation} \begin{equation}=\frac{x}{2\cdot \left ( x-4 \right )} \end{equation} 3)\begin{equation}\frac{1}{2}-\frac{x}{2\cdot \left ( x-4 \right )}= \end{equation} \begin{equation}=\frac{x-4-x}{2\cdot \left ( x-4 \right )}= \end{equation} \begin{equation}=-\frac{4}{2\left ( x-4 \right )}=-\frac{2}{x-4} \end{equation} 4)\begin{equation}\left ( \frac{3}{\left ( 3b-a \right )^{2}}-\frac{1}{a^{2}-9b^{2}} \right ):\frac{a+6b}{a^{2}b-9b^{3}}-\frac{a-b}{a-3b} \end{equation} 1)\begin{equation}\frac{3}{\left ( 3b-a \right )^{2}}-\frac{1}{a^{2}-9b^{2}}= \end{equation} \begin{equation}=\frac{3a+9b-a+3b}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )}= \end{equation} \begin{equation}=\frac{2a+12b}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )}= \end{equation} \begin{equation}=\frac{2\left ( a+6b \right )}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )} \end{equation} 2)\begin{equation}\frac{2\cdot \left ( a+6b \right )}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )}:\frac{a+6b}{a^{2}b-9b^{3}}= \end{equation} \begin{equation}=\frac{2\cdot \left ( a+6b \right )}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )}\cdot \frac{b\left ( a^{2}-9b^{2} \right )}{a+6b}= \end{equation} \begin{equation}=\frac{2\cdot b\cdot \left ( a-3b \right )\left ( a+3b \right )}{\left ( 3b-a \right )^{2}\cdot \left ( a+3b \right )}= \end{equation} \begin{equation}=\frac{2b}{3b-a} \end{equation} 3)\begin{equation}\frac{2b}{3b-a}-\frac{a-b}{a-3b}= \end{equation} \begin{equation} =\frac{2b}{3b-a}+\frac{a-b}{3b-a} \end{equation} 5)\begin{equation}\left ( \frac{a-b}{ab}-\frac{3a+b}{ab-b^{2}}+\frac{3b+a}{ab-a^{2}} \right ):\frac{2a+2b}{ab}-\frac{2a}{b-a} \end{equation} 1)\begin{equation}\frac{a-b}{ab}-\frac{3a+b}{ab-b^{2}}+\frac{3b+a}{ab-a^{2}}= \end{equation} \begin{equation}=\frac{a-b}{ab}-\frac{3a+b}{b\left ( a-b \right )}+\frac{3b+a}{a\left ( b-a \right )}= \end{equation} \begin{equation}=\frac{a-b}{ab}-\frac{3a+b}{b\left ( a-b \right )}-\frac{3b+a}{a\left ( a-b \right )}= \end{equation} \begin{equation}=\frac{a^{2}-2ab+b^{2}-3a^{2}-ab-3b^{2}-ab}{ab\left ( a-b \right )}= \end{equation} \begin{equation}=\frac{-2a^{2}-4ab-2b^{2}}{ab\left ( a-b \right )}= \end{equation} \begin{equation}=\frac{-2\left ( a^{2}+2ab+b^{2} \right )}{ab\left ( a-b \right )}= \end{equation} \begin{equation}=\frac{-2\left ( a+b \right )^{2}}{ab\left ( a-b \right )} \end{equation} 2)\begin{equation}\frac{-2\left ( a+b \right )^{2}}{ab\left ( a-b \right )}:\frac{2a+2b}{ab}= \end{equation} \begin{equation}=\frac{-2\left ( a+b \right )^{2}}{ab\left ( a-b \right )}\cdot \frac{ab}{2\left ( a+b \right )}= \end{equation} \begin{equation}=-\frac{\left ( a+b \right )}{\left ( a-b \right )} \end{equation} 3)\begin{equation}-\frac{\left ( a+b \right )}{\left ( a-b \right )}-\frac{2a}{b-a}= \end{equation} \begin{equation}=\frac{-a-b}{a-b}+\frac{2a}{a-b}= \end{equation} \begin{equation}=\frac{-a-b+2a}{a-b}=\frac{a-b}{a-b}=1 \end{equation} 6)\begin{equation}\frac{b}{x-b}-\frac{ab}{x-a}\cdot \left ( \frac{x+a}{ax-ab}+\frac{x+b}{b^{2}-bx}-\frac{x}{ab} \right ) \end{equation} 1)\begin{equation}\frac{x+a}{ax-ab}+\frac{x+b}{b^{2}-bx}-\frac{x}{ab}= \end{equation} \begin{equation}\frac{x+a}{a\left ( x-b \right )}+\frac{x+b}{b\left ( b-x \right )}-\frac{x}{ab}= \end{equation} \begin{equation}=\frac{x+a}{a\left ( x-b \right )}-\frac{x+b}{b\left ( x-b \right )}-\frac{x}{ab}= \end{equation} \begin{equation}=\frac{bx+ab-ax-ab-x^{2}+bx}{ab\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{bx-ax-x^{2}+bx}{ab\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{x\left ( b-a-x+b \right )}{ab\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{x\left ( 2b-a-x \right )}{ab\left ( x-b \right )} \end{equation} 2)\begin{equation}\frac{ab}{x-a}\cdot \frac{x\left ( 2b-a-x \right )}{ab\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{x\left ( 2b-a-x \right )}{\left ( x-a \right )\left ( x-b \right )} \end{equation} 3)\begin{equation}\frac{b}{x-b}-\frac{x\left ( 2b-a-x \right )}{\left ( x-a \right )\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{bx-ab-2bx+ax+x^{2}}{\left ( x-a \right )\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{-bx-ab+ax+x^{2}}{\left ( x-a \right )\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{-b\left ( x+a \right )+x\left ( a+x \right )}{\left ( x-a \right )\left ( x-b \right )}= \end{equation} \begin{equation}=\frac{\left ( x+a \right )\left ( x-b \right )}{\left ( x-a \right )\left ( x-b \right )}=\frac{x+a}{x-a} \end{equation} 7)\begin{equation}\left ( a+\frac{a-b}{a+b}-b \right ):\left ( \frac{2a+1}{a^{2}-b^{2}}+1 \right ) \end{equation} 1)\begin{equation}a+\frac{a-b}{a+b}-b= \end{equation} \begin{equation}=\frac{a^{2}+ab+a-b-ab-b^{2}}{a+b}= \end{equation} \begin{equation}=\frac{\left ( a^{2}-b^{2} \right )+\left ( a-b \right )}{a+b}= \end{equation} \begin{equation}=\frac{\left ( a-b \right )\left ( a+b \right )+\left ( a-b \right )}{a+b}= \end{equation} \begin{equation}=\frac{\left ( a-b \right )\left ( a+b+1 \right )}{a+b} \end{equation} 2)\begin{equation}\frac{2a+1}{a^{2}-b^{2}}+1=\frac{2a+1+a^{2}-b^{2}}{a^{2}-b^{2}} \end{equation} 3)\begin{equation}\frac{\left ( a-b \right )\left ( a+b+1 \right )}{a+b}:\frac{2a+1+a^{2}-b^{2}}{a^{2}-b^{2}}= \end{equation} \begin{equation}=\frac{\left ( a-b \right )\left ( a+b+1 \right )}{a+b}\cdot \frac{\left ( a+b \right )\left ( a-b \right )}{\left ( 2a+1+a^{2}-b^{2} \right )}= \end{equation} \begin{equation}=\frac{\left ( a-b \right )^{2}\left ( a+b+1 \right )}{2a+1+a^{2}-b^{2}} \end{equation} 8)\begin{equation}\left ( x-\frac{x+y}{x-y}+y \right ):\left ( 1-\frac{2y+1}{x^{2}-y^{2}} \right ) \end{equation} 1)\begin{equation}x-\frac{x+y}{x-y}+y=\left ( x+y \right )-\frac{x+y}{x-y}= \end{equation} \begin{equation}=\frac{x^{2}-y^{2}-\left ( x+y \right )}{x-y}= \end{equation} \begin{equation}=\frac{\left ( x+y \right )\left ( x-y \right )-\left ( x+y \right )}{x-y}= \end{equation} \begin{equation}=\frac{\left ( x+y \right )\left ( x-y-1 \right )}{x-y} \end{equation} 2)\begin{equation}1-\frac{2y+1}{x^{2}-y^{2}}=\frac{x^{2}-y^{2}-2y-1}{x^{2}-y^{2}} \end{equation} 3)\begin{equation}\frac{\left ( x+y \right )\left ( x-y-1 \right )}{x-y}:\frac{x^{2}-y^{2}}{x^{2}-y^{2}-2y-1}= \end{equation} \begin{equation}=\frac{\left ( x+y \right )\left ( x-y-1 \right )}{x-y}\cdot \frac{\left ( x-y \right )\left ( x+y \right )}{x^{2}-y^{2}-2y-1}= \end{equation} \begin{equation}=\frac{\left ( x+y \right )^{2}\left ( x-y-1 \right )}{x^{2}-y^{2}-2y-1} \end{equation} \begin{equation} \end{equation}
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