Solve Linear Equation 8B0Aa9

1. **State the problem:** Solve the equation $$\frac{4}{5} x + \frac{6}{7} = 6x + \frac{97}{3}$$ for $x$. 2. **Write down the equation:** $$\frac{4}{5} x + \frac{6}{7} = 6x + \frac{97}{3}$$ 3. **Goal:** Isolate $x$ on one side. 4. **Subtract $6x$ from both sides:** $$\frac{4}{5} x - 6x + \frac{6}{7} = \frac{97}{3}$$ 5. **Combine like terms for $x$:** $$\frac{4}{5} x - 6x = \frac{4}{5} x - \frac{30}{5} x = \left(\frac{4}{5} - \frac{30}{5}\right) x = -\frac{26}{5} x$$ So the equation becomes: $$-\frac{26}{5} x + \frac{6}{7} = \frac{97}{3}$$ 6. **Subtract $\frac{6}{7}$ from both sides:** $$-\frac{26}{5} x = \frac{97}{3} - \frac{6}{7}$$ 7. **Find common denominator for right side:** The denominators are $3$ and $7$, common denominator is $21$. $$\frac{97}{3} = \frac{97 \times 7}{21} = \frac{679}{21}$$ $$\frac{6}{7} = \frac{6 \times 3}{21} = \frac{18}{21}$$ So, $$\frac{97}{3} - \frac{6}{7} = \frac{679}{21} - \frac{18}{21} = \frac{661}{21}$$ 8. **Rewrite equation:** $$-\frac{26}{5} x = \frac{661}{21}$$ 9. **Divide both sides by $-\frac{26}{5}$:** $$x = \frac{\frac{661}{21}}{-\frac{26}{5}} = \frac{661}{21} \times \frac{5}{-26} = - \frac{661 \times 5}{21 \times 26}$$ 10. **Calculate numerator and denominator:** $$661 \times 5 = 3305$$ $$21 \times 26 = 546$$ So, $$x = - \frac{3305}{546}$$ 11. **Simplify fraction:** Find gcd of $3305$ and $546$. - $3305 = 5 \times 661$ - $546 = 2 \times 3 \times 7 \times 13$ No common factors, so fraction is in simplest form. **Final answer:** $$x = - \frac{3305}{546}$$