1. **Stating the problem:** Solve the equation for 401: $ (2x - 1)^2 - (2 - x)(1 + 3x) = (x - 1)(7x - 2) $. 2. **Formula and rules:** We will expand all products using distributive property and then simplify both sides. 3. **Expand each term:** $$ (2x - 1)^2 = (2x)^2 - 2 \times 2x \times 1 + 1^2 = 4x^2 - 4x + 1 $$ $$ (2 - x)(1 + 3x) = 2 \times 1 + 2 \times 3x - x \times 1 - x \times 3x = 2 + 6x - x - 3x^2 = 2 + 5x - 3x^2 $$ $$ (x - 1)(7x - 2) = x \times 7x - x \times 2 - 1 \times 7x + 1 \times 2 = 7x^2 - 2x - 7x + 2 = 7x^2 - 9x + 2 $$ 4. **Rewrite the equation:** $$ 4x^2 - 4x + 1 - (2 + 5x - 3x^2) = 7x^2 - 9x + 2 $$ 5. **Distribute the minus sign:** $$ 4x^2 - 4x + 1 - 2 - 5x + 3x^2 = 7x^2 - 9x + 2 $$ 6. **Combine like terms on the left:** $$ (4x^2 + 3x^2) + (-4x - 5x) + (1 - 2) = 7x^2 - 9x + 2 $$ $$ 7x^2 - 9x - 1 = 7x^2 - 9x + 2 $$ 7. **Subtract $7x^2 - 9x$ from both sides:** $$ \cancel{7x^2} - \cancel{9x} - 1 = \cancel{7x^2} - \cancel{9x} + 2 $$ $$ -1 = 2 $$ 8. **Conclusion:** The statement $-1 = 2$ is false, so there is no solution to the equation. --- 1. **Stating the problem:** Solve the equation for 402: $ \frac{1}{2} x - \frac{x + 3}{10} + \frac{x}{5} = \frac{3}{5} x - \frac{3}{10} $. 2. **Formula and rules:** We will clear denominators by multiplying through by the least common denominator (LCD) and then simplify. 3. **Identify LCD:** Denominators are 2, 10, 5, 5, 10. LCD is 10. 4. **Multiply entire equation by 10:** $$ 10 \times \left( \frac{1}{2} x - \frac{x + 3}{10} + \frac{x}{5} \right) = 10 \times \left( \frac{3}{5} x - \frac{3}{10} \right) $$ 5. **Distribute multiplication:** $$ 10 \times \frac{1}{2} x = 5x $$ $$ 10 \times \frac{x + 3}{10} = x + 3 $$ $$ 10 \times \frac{x}{5} = 2x $$ $$ 10 \times \frac{3}{5} x = 6x $$ $$ 10 \times \frac{3}{10} = 3 $$ 6. **Rewrite equation:** $$ 5x - (x + 3) + 2x = 6x - 3 $$ 7. **Distribute minus sign:** $$ 5x - x - 3 + 2x = 6x - 3 $$ 8. **Combine like terms on left:** $$ (5x - x + 2x) - 3 = 6x - 3 $$ $$ 6x - 3 = 6x - 3 $$ 9. **Subtract $6x$ from both sides:** $$ \cancel{6x} - 3 = \cancel{6x} - 3 $$ $$ -3 = -3 $$ 10. **Conclusion:** The equation is true for all $x$, so the solution is all real numbers.