1. **Problem 1:** Find the equation of a line passing through $(-2,5)$ and parallel to the line $6y + 2x = 10$. 2. **Step 1:** Rewrite the given line in slope-intercept form $y = mx + b$ to find its slope. $$6y + 2x = 10 \implies 6y = -2x + 10 \implies y = -\frac{2}{6}x + \frac{10}{6} = -\frac{1}{3}x + \frac{5}{3}$$ The slope $m$ of the given line is $-\frac{1}{3}$. 3. **Step 2:** Since parallel lines have the same slope, the new line also has slope $m = -\frac{1}{3}$. 4. **Step 3:** Use point-slope form of a line equation: $$y - y_1 = m(x - x_1)$$ where $(x_1,y_1) = (-2,5)$ and $m = -\frac{1}{3}$. 5. **Step 4:** Substitute values: $$y - 5 = -\frac{1}{3}(x + 2)$$ 6. **Step 5:** Simplify: $$y - 5 = -\frac{1}{3}x - \frac{2}{3} \implies y = -\frac{1}{3}x - \frac{2}{3} + 5 = -\frac{1}{3}x + \frac{13}{3}$$ --- 7. **Problem 2:** Calculate the length of a rectangular air pipe made from 63 kg of metal with density 8000 kg/m³. External dimensions: 12 cm by 15 cm Internal dimensions: $l$ cm by 12 cm 8. **Step 1:** Convert dimensions to meters: External: $0.12$ m by $0.15$ m Internal: $\frac{l}{100}$ m by $0.12$ m 9. **Step 2:** Volume of metal used = volume of external pipe - volume of internal pipe $$V = 63 / 8000 = 0.007875 \text{ m}^3$$ 10. **Step 3:** Volume formula: $$V = \text{length} \times (\text{external area} - \text{internal area})$$ $$0.007875 = L \times (0.12 \times 0.15 - \frac{l}{100} \times 0.12)$$ $$0.007875 = L \times (0.018 - 0.0012l)$$ 11. **Step 4:** Solve for $L$: $$L = \frac{0.007875}{0.018 - 0.0012l}$$ Length $L$ in meters depends on $l$. --- 12. **Problem 3:** Find the equation of a line passing through $(5,9)$ and parallel to $5y + 2x = 10$. 13. **Step 1:** Rewrite given line in slope-intercept form: $$5y + 2x = 10 \implies 5y = -2x + 10 \implies y = -\frac{2}{5}x + 2$$ Slope $m = -\frac{2}{5}$. 14. **Step 2:** Use point-slope form with point $(5,9)$: $$y - 9 = -\frac{2}{5}(x - 5)$$ 15. **Step 3:** Simplify: $$y - 9 = -\frac{2}{5}x + 2 \implies y = -\frac{2}{5}x + 11$$