1. Problem a: Find parametric equations for the solution set of $7x - 5y = 3$. 2. This is a linear equation in two variables. We can express one variable in terms of the other. 3. Solve for $x$: $$7x = 3 + 5y$$ $$x = \frac{3 + 5y}{7}$$ 4. Let $y = t$ (parameter). Then: $$x = \frac{3 + 5t}{7}$$ 5. Parametric form: $$x = \frac{3 + 5t}{7}, \quad y = t$$ --- 6. Problem b: Find parametric equations for $3x_1 - 5x_2 + 4x_3 = 7$. 7. This is a linear equation in three variables. We can express one variable in terms of the other two. 8. Solve for $x_1$: $$3x_1 = 7 + 5x_2 - 4x_3$$ $$x_1 = \frac{7 + 5x_2 - 4x_3}{3}$$ 9. Let $x_2 = s$, $x_3 = t$ (parameters). Then: $$x_1 = \frac{7 + 5s - 4t}{3}, \quad x_2 = s, \quad x_3 = t$$ --- 10. Problem c: Find parametric equations for $-8x_1 + 2x_2 - 5x_3 + 6x_4 = 1$. 11. Solve for $x_1$: $$-8x_1 = 1 - 2x_2 + 5x_3 - 6x_4$$ $$x_1 = \frac{-1 + 2x_2 - 5x_3 + 6x_4}{8}$$ 12. Let $x_2 = s$, $x_3 = t$, $x_4 = u$ (parameters). Then: $$x_1 = \frac{-1 + 2s - 5t + 6u}{8}, \quad x_2 = s, \quad x_3 = t, \quad x_4 = u$$ --- 13. Problem d: Find parametric equations for $3v - 8w + 2x - y + 4z = 0$. 14. Solve for $v$: $$3v = 8w - 2x + y - 4z$$ $$v = \frac{8w - 2x + y - 4z}{3}$$ 15. Let $w = a$, $x = b$, $y = c$, $z = d$ (parameters). Then: $$v = \frac{8a - 2b + c - 4d}{3}, \quad w = a, \quad x = b, \quad y = c, \quad z = d$$