Solve the equation: $$13\frac{17}{35} - x = 20 - 13\frac{27}{28}$$

Let's solve the equation step by step. First, we have the equation: $$13\frac{17}{35} - x = 20 - 13\frac{27}{28}$$ To isolate $$x$$, we can rewrite the equation as: $$x = 13\frac{17}{35} - (20 - 13\frac{27}{28})$$ $$x = 13\frac{17}{35} - 20 + 13\frac{27}{28}$$ Now, let's rewrite the mixed numbers as improper fractions: $$13\frac{17}{35} = \frac{13 \cdot 35 + 17}{35} = \frac{455 + 17}{35} = \frac{472}{35}$$ $$13\frac{27}{28} = \frac{13 \cdot 28 + 27}{28} = \frac{364 + 27}{28} = \frac{391}{28}$$ Substitute these improper fractions back into the equation: $$x = \frac{472}{35} - 20 + \frac{391}{28}$$ To combine these terms, we need a common denominator for the fractions. The least common multiple (LCM) of 35 and 28 is 140. So, we convert the fractions to have a denominator of 140: $$\frac{472}{35} = \frac{472 \cdot 4}{35 \cdot 4} = \frac{1888}{140}$$ $$\frac{391}{28} = \frac{391 \cdot 5}{28 \cdot 5} = \frac{1955}{140}$$ And rewrite 20 with the same denominator: $$20 = \frac{20 \cdot 140}{140} = \frac{2800}{140}$$ Now we have: $$x = \frac{1888}{140} - \frac{2800}{140} + \frac{1955}{140}$$ $$x = \frac{1888 - 2800 + 1955}{140}$$ $$x = \frac{-912 + 1955}{140}$$ $$x = \frac{1043}{140}$$ Now, convert the improper fraction back to a mixed number: $$x = \frac{1043}{140} = 7\frac{63}{140}$$ We can simplify the fraction $$\frac{63}{140}$$ by dividing both numerator and denominator by 7: $$\frac{63}{140} = \frac{63 \div 7}{140 \div 7} = \frac{9}{20}$$ So, the final answer is: $$x = 7\frac{9}{20}$$ Answer: $$7\frac{9}{20}$$