Solve Linear Equation 614Ad7

1. **State the problem:** Solve the equation $$-\frac{6}{2} \left(5 + \frac{x}{3}\right) = \frac{1}{3} \left(4 - \frac{4x}{2}\right) + \frac{3x}{2}$$ for $x$. 2. **Simplify the coefficients:** $$-\frac{6}{2} = -3$$ $$-\frac{4x}{2} = -2x$$ Rewrite the equation: $$-3 \left(5 + \frac{x}{3}\right) = \frac{1}{3} (4 - 2x) + \frac{3x}{2}$$ 3. **Distribute on the left side:** $$-3 \times 5 = -15$$ $$-3 \times \frac{x}{3} = -x$$ So left side becomes: $$-15 - x$$ 4. **Distribute on the right side:** $$\frac{1}{3} \times 4 = \frac{4}{3}$$ $$\frac{1}{3} \times (-2x) = -\frac{2x}{3}$$ Right side is: $$\frac{4}{3} - \frac{2x}{3} + \frac{3x}{2}$$ 5. **Combine like terms on the right side:** Find common denominator for $-\frac{2x}{3}$ and $\frac{3x}{2}$, which is 6. Rewrite: $$-\frac{2x}{3} = -\frac{4x}{6}$$ $$\frac{3x}{2} = \frac{9x}{6}$$ Sum: $$-\frac{4x}{6} + \frac{9x}{6} = \frac{5x}{6}$$ So right side is: $$\frac{4}{3} + \frac{5x}{6}$$ 6. **Rewrite the equation:** $$-15 - x = \frac{4}{3} + \frac{5x}{6}$$ 7. **Move all $x$ terms to one side and constants to the other:** Add $x$ to both sides: $$-15 = \frac{4}{3} + \frac{5x}{6} + x$$ Rewrite $x$ as $\frac{6x}{6}$: $$-15 = \frac{4}{3} + \frac{5x}{6} + \frac{6x}{6} = \frac{4}{3} + \frac{11x}{6}$$ Subtract $\frac{4}{3}$ from both sides: $$-15 - \frac{4}{3} = \frac{11x}{6}$$ 8. **Simplify left side:** Convert $-15$ to thirds: $$-15 = -\frac{45}{3}$$ So: $$-\frac{45}{3} - \frac{4}{3} = -\frac{49}{3}$$ Equation is: $$-\frac{49}{3} = \frac{11x}{6}$$ 9. **Solve for $x$ by multiplying both sides by $\frac{6}{11}$:** $$x = -\frac{49}{3} \times \frac{6}{11}$$ 10. **Simplify:** $$x = -\frac{49 \times 6}{3 \times 11} = -\frac{294}{33}$$ Divide numerator and denominator by 3: $$x = -\frac{98}{11}$$ 11. **Final answer:** $$x = -\frac{98}{11}$$ **Note:** The user provided $x = -\frac{38}{11}$, but the correct solution is $x = -\frac{98}{11}$ based on the steps above.