Đáp án:
$a)
a+b-c=1+3-2=2$
b)
Vậy phân số cần tìm là $\left ( \dfrac{1}{a};\dfrac{1}{b} \right )=\left \{\left ( \dfrac{1}{-4};\dfrac{1}{1} \right ),\left ( \dfrac{1}{5};\dfrac{1}{4} \right );\left (\dfrac{1}{8};\dfrac{1}{3} \right ) \right\}$
Giải thích các bước giải:
$a)
\dfrac{52}{9}=5+\dfrac{1}{\dfrac{9}{7}}\\
\dfrac{9}{7}=1+\dfrac{1}{\dfrac{7}{2}}\\
\dfrac{7}{2}=3+\dfrac{1}{2}\\
\Rightarrow \dfrac{52}{9}=5+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{2}}}\\
\Rightarrow a=1,b=3,c=2\\
\Rightarrow a+b-c=1+3-2=2$
b)
Gọi hai phân số cần tìm có dạng $\dfrac{1}{a},\dfrac{1}{b}(a,b \neq 0)$
Ta có
$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{ab}=\dfrac{1}{2}\\
\Leftrightarrow \dfrac{b}{ab}+\dfrac{a}{ab}+\dfrac{1}{ab}=\dfrac{1}{2}\\
\Leftrightarrow \dfrac{a+b+1}{ab}=\dfrac{1}{2}\\
\Leftrightarrow 2a+2b+2=ab\\
\Leftrightarrow 2a+2b+2-ab=0\\
\Leftrightarrow a(2-b)-2(2-b)+6=0\\
\Leftrightarrow (2-b)(a-2)=-6\\
\Leftrightarrow {\left[\begin{aligned}{\left\{\begin{aligned} 2-b=-1\\a-2=6\end{aligned}\right.}\\ {\left\{\begin{aligned} 2-b=1\\a-2=-6 \end{aligned}\right.}\\ {\left\{\begin{aligned} 2-b=2\\a-2=-3 \end{aligned}\right.}\\ {\left\{\begin{aligned} 2-b=-2\\a-2=3 \end{aligned}\right.} \end{aligned}\right.}\\
\Leftrightarrow {\left[\begin{aligned}{\left\{\begin{aligned} b=3\\a=8\end{aligned}\right.}\\ {\left\{\begin{aligned} b=1\\a=-4 \end{aligned}\right.}\\ {\left\{\begin{aligned} b=0(KTMĐK)\\a=-1 \end{aligned}\right.}\\ {\left\{\begin{aligned} b=4\\a=5 \end{aligned}\right.} \end{aligned}\right.}$
Vậy phân số cần tìm là $\left ( \dfrac{1}{a};\dfrac{1}{b} \right )=\left\{\left ( \dfrac{1}{-4};\dfrac{1}{1} \right ),\left ( \dfrac{1}{5};\dfrac{1}{4} \right );\left (\dfrac{1}{8};\dfrac{1}{3} \right ) \right\}$
