1. Problem 9(a): Find the solution set of the linear equation $$2x + 4y = 3$$. Step 1: Write the equation $$2x + 4y = 3$$. Step 2: Solve for one variable in terms of the other. Let's solve for $$x$$: $$2x = 3 - 4y$$ $$x = \frac{3}{2} - 2y$$ Step 3: Introduce parameter $$t$$ to represent $$y$$: Let $$y = t$$, where $$t \in \mathbb{R}$$. Step 4: Express the solution set as: $$\left(x,y\right) = \left( \frac{3}{2} - 2t, t \right), t \in \mathbb{R}$$. 2. Problem 9(b): Find the solution set of $$3x_1 - 5x_2 + x_3 + 4x_4 = 9$$. Step 1: The equation is $$3x_1 - 5x_2 + x_3 + 4x_4 = 9$$. Step 2: Solve for $$x_1$$: $$3x_1 = 9 + 5x_2 - x_3 - 4x_4$$ $$x_1 = 3 + \frac{5}{3}x_2 - \frac{1}{3}x_3 - \frac{4}{3}x_4$$ Step 3: Introduce parameters: Let $$x_2 = s$$, $$x_3 = t$$, $$x_4 = u$$ where $$s,t,u \in \mathbb{R}$$. Step 4: The solution set is: $$\left(x_1,x_2,x_3,x_4\right) = \left(3 + \frac{5}{3}s - \frac{1}{3}t - \frac{4}{3}u, s, t, u\right), s,t,u \in \mathbb{R}$$. 3. Problem 10(a): Find the solution set of $$3x_1 - 5x_2 + 4x_3 = 7$$. Step 1: Given $$3x_1 - 5x_2 + 4x_3 = 7$$. Step 2: Solve for $$x_1$$: $$3x_1 = 7 + 5x_2 - 4x_3$$ $$x_1 = \frac{7}{3} + \frac{5}{3}x_2 - \frac{4}{3}x_3$$ Step 3: Let parameters be: $$x_2 = s, x_3 = t, s,t \in \mathbb{R}$$. Step 4: Solution set: $$\left(x_1,x_2,x_3\right) = \left(\frac{7}{3} + \frac{5}{3}s - \frac{4}{3}t, s, t\right), s,t \in \mathbb{R}$$. 4. Problem 10(b): Find solution set for $$3v - 8w + 2r - y + 4z = 0$$. Step 1: Equation is $$3v - 8w + 2r - y + 4z = 0$$. Step 2: Solve for $$v$$: $$3v = 8w - 2r + y - 4z$$ $$v = \frac{8}{3}w - \frac{2}{3}r + \frac{1}{3}y - \frac{4}{3}z$$ Step 3: Parameters: Let $$w = a, r = b, y = c, z = d, a,b,c,d \in \mathbb{R}$$. Step 4: Solution set: $$\left(v,w,r,y,z\right) = \left(\frac{8}{3}a - \frac{2}{3}b + \frac{1}{3}c - \frac{4}{3}d, a, b, c, d\right), a,b,c,d \in \mathbb{R}$$. 5. Problem 11: Construct systems of linear equations from the augmented matrices given by points (a)-(e): (a) Coordinates: $\left(\frac{5}{7}, \frac{8}{7}, 1\right)$ Equation form: $ax + by = c$, so the system is $$\frac{5}{7}x + \frac{8}{7}y = 1$$ Rewrite: $$5x + 8y = 7$$ (b) Coordinates: $\left(\frac{5}{7}, \frac{8}{7}, 0\right)$ System: $$\frac{5}{7}x + \frac{8}{7}y = 0$$ Rewrite: $$5x + 8y = 0$$ (c) $\left(5, 8, 1\right)$ System: $$5x + 8y = 1$$ (d) $\left(\frac{5}{7}, \frac{10}{7}, \frac{2}{7}\right)$ Equation: $$\frac{5}{7}x + \frac{10}{7}y = \frac{2}{7}$$ Multiply both sides by 7: $$5x + 10y = 2$$ (e) $\left(\frac{5}{7}, \frac{22}{7}, 2\right)$ Equation: $$\frac{5}{7}x + \frac{22}{7}y = 2$$ Multiply both sides by 7: $$5x + 22y = 14$$ Each represents a linear equation system with variables $$x,y$$, corresponding to the augmented matrix entries. Final answers are the parameterized solution sets for Problems 9 and 10, and corresponding equations for 11 as above.