Problem S:E25.67

         In tranlation

                              " S:E25.67. Knowing that $x+\frac{1}{x}=7$,  calculate $\frac{1}{x}-\frac{1}{x-7}.$"
 

ANSWER CiP

$$\frac{1}{x}-\frac{1}{x-7}=7$$

                            Solution CiP

               $x+\frac{1}{x}=7\;\Leftrightarrow\;x^2-7x+1=0 \tag{1}$

Now

$$\frac{1}{x}-\frac{1}{x-7}\underset{(1)}{=}7-x-\frac{1}{x-7}=7-\frac{x^2-7x+1}{x-7}\underset{(1)}{=}7-\frac{0}{x-7}=7$$

$\blacksquare$

           Remark CiP 

          The question is whether there are other expressions like this, $\frac{1}{x+m}+\frac{n}{px+q}$  which can be calculated from the given condition $x+\frac{1}{x}=7.$