Solve for X: Complex Fraction Equation with 1/7, 3/5x, and 1/8x Terms

To solve the equation $\frac{1}{7} - \frac{3}{5}x + \frac{1}{8}x = \frac{1}{9} + \frac{3}{9} - \frac{2}{10}x$, follow these steps:

• Step 1: Simplify constant terms
  Combine the constant terms on the right side: $\frac{1}{9} + \frac{3}{9} = \frac{4}{9}$.
• Step 2: Handle fractions involving $x$ and simplify
  On the left: Combine $-\frac{3}{5}x$ and $\frac{1}{8}x$ to get a single fraction with common denominator 40: $-\frac{3}{5} \times \frac{8}{8}x + \frac{1}{8} \times \frac{5}{5}x = -\frac{24}{40}x + \frac{5}{40}x = -\frac{19}{40}x$.
• Step 3: Isolate terms involving $x$
  Rewrite the equation: $\frac{1}{7} - \frac{19}{40}x = \frac{4}{9} - \frac{2}{10}x$.
  Bring all $x$-terms to the left, and constant terms to the right: $\frac{1}{7} - \frac{4}{9} = -\frac{2}{10}x + \frac{19}{40}x$.
• Step 4: Simplify each side
  For the constants, find a common denominator 63: $\frac{1}{7} \times \frac{9}{9} - \frac{4}{9} \times \frac{7}{7} = \frac{9}{63} - \frac{28}{63} = \frac{-19}{63}$.
  For $x$-terms, common denominator 40: $-\frac{8}{40}x + \frac{19}{40}x = \frac{11}{40}x$.
• Step 5: Solve for $x$
  Combine: $\frac{-19}{63} = \frac{11}{40}x$.
  Solve: $x = \frac{-19}{63} \times \frac{40}{11} = \frac{-760}{693}$.

Therefore, the solution to the problem is $x = -\frac{760}{693}$.