1. **Problem:** Solve the first-degree equation analytically and graphically: $$3 \cdot \left( \frac{2x - 1}{3} - \frac{1 - 2x}{2} \right) = \frac{2 - x}{3} - 1 + x$$ 2. **Formula and rules:** To solve linear equations, we use the distributive property, combine like terms, and isolate $x$. 3. **Step 1:** Simplify inside the parentheses: $$\frac{2x - 1}{3} - \frac{1 - 2x}{2} = \frac{2(2x - 1)}{6} - \frac{3(1 - 2x)}{6} = \frac{4x - 2 - 3 + 6x}{6} = \frac{10x - 5}{6}$$ 4. **Step 2:** Multiply by 3: $$3 \cdot \frac{10x - 5}{6} = \frac{3(10x - 5)}{6} = \frac{30x - 15}{6} = \frac{30x}{6} - \frac{15}{6} = 5x - \frac{5}{2}$$ 5. **Step 3:** Simplify the right side: $$\frac{2 - x}{3} - 1 + x = \frac{2 - x}{3} - \frac{3}{3} + \frac{3x}{3} = \frac{2 - x - 3 + 3x}{3} = \frac{-1 + 2x}{3}$$ 6. **Step 4:** Set the equation: $$5x - \frac{5}{2} = \frac{-1 + 2x}{3}$$ 7. **Step 5:** Multiply both sides by 6 (common denominator) to clear fractions: $$6 \cdot \left(5x - \frac{5}{2}\right) = 6 \cdot \frac{-1 + 2x}{3}$$ $$6 \cdot 5x - 6 \cdot \frac{5}{2} = 2 \cdot (-1 + 2x)$$ $$30x - 15 = -2 + 4x$$ 8. **Step 6:** Bring all terms to one side: $$30x - 15 - 4x + 2 = 0$$ $$26x - 13 = 0$$ 9. **Step 7:** Solve for $x$: $$26x = 13$$ $$\cancel{26}x = \cancel{13}$$ $$x = \frac{13}{26} = \frac{1}{2}$$ **Final answer:** $x = \frac{1}{2}$ --- **Slug:** linear-equation **Subject:** algebra **svg:** "" **desmos:** {"latex": "y=3 \cdot \left( \frac{2x - 1}{3} - \frac{1 - 2x}{2} \right) - \left( \frac{2 - x}{3} - 1 + x \right)", "features": {"intercepts": true, "extrema": true}} **q_count:** 2