Solve Linear Equation 82C6D1

1. **State the problem:** Solve the equation $$\frac{4}{5} x + \frac{6}{7} = 7x + \frac{87}{4}$$ for $x$. 2. **Write down the equation:** $$\frac{4}{5} x + \frac{6}{7} = 7x + \frac{87}{4}$$ 3. **Goal:** Isolate $x$ on one side. 4. **Subtract $7x$ from both sides:** $$\frac{4}{5} x - 7x + \frac{6}{7} = \frac{87}{4}$$ 5. **Combine like terms for $x$:** $$\frac{4}{5} x - 7x = \frac{4}{5} x - \frac{35}{5} x = \frac{4 - 35}{5} x = \frac{-31}{5} x$$ So the equation becomes: $$\frac{-31}{5} x + \frac{6}{7} = \frac{87}{4}$$ 6. **Subtract $\frac{6}{7}$ from both sides:** $$\frac{-31}{5} x = \frac{87}{4} - \frac{6}{7}$$ 7. **Find common denominator for right side:** Denominators are $4$ and $7$, common denominator is $28$. $$\frac{87}{4} = \frac{87 \times 7}{28} = \frac{609}{28}$$ $$\frac{6}{7} = \frac{6 \times 4}{28} = \frac{24}{28}$$ So: $$\frac{87}{4} - \frac{6}{7} = \frac{609}{28} - \frac{24}{28} = \frac{585}{28}$$ 8. **Rewrite equation:** $$\frac{-31}{5} x = \frac{585}{28}$$ 9. **Solve for $x$ by dividing both sides by $\frac{-31}{5}$:** $$x = \frac{585}{28} \div \frac{-31}{5} = \frac{585}{28} \times \frac{5}{-31}$$ 10. **Multiply numerators and denominators:** $$x = \frac{585 \times 5}{28 \times (-31)} = \frac{2925}{-868}$$ 11. **Simplify fraction by dividing numerator and denominator by 29:** $$\frac{2925 \div 29}{-868 \div 29} = \frac{101}{-30} = -\frac{101}{30}$$ **Final answer:** $$x = -\frac{101}{30}$$