Exercise 1.3.5 For each of the following homogeneous systems, find a set of basic solutions and express the gen- eral solution as a linear combination of these basic solu- tions. a. b. - x1 + 2x2 x3 + 2x4+x5=0 x1 + 2x2 + 2x3 +x5=0 2x1 + 4x2 - 2x3 + 3x4+x5=0 - x1 + 2x2 xX3+x4+ x5 = 0 -x1-2x2+2x3 + x5=0 -x1-2x2+3x3 + x4+3x5=0 c. - x1 + x2 x3 + 2x4 + x5 = 0 x5=0 x5 = 0 x1 + 2x2 x3 + x4+ - 2x13x2 x3+2x4 + x5=0 - 4x15x22x3 +5x4+2x5 = 0

Express every solution of the following system as a sum of a specific solution plus a solution of the homogeneous system:

For two linear equations in three unknowns x, y, z, the row picture will show (2 or 3) (lines or planes) in (2 or 3)-dimensional space. The column picture is in (2 or 3)dimensional space. The solutions normally lie on a __ .

Find the solution set of the system of linear equations over Z5: +x2 +x3 X1 X1 2x1 +3x2 +x2 +x3 +x4 +2x4 = 2 3 1

Evaluate the derived system of Linear equations from a given nodal analysis circuit using Gauss-Seidel Method. Tabulate the results and use a terminating condition of Ea <0.00001 for each variable. 0 = 5 I1 – 9 I3 + 3 I2 -8 = 2 I3 – 5 I2 + 3 I1 16 = 5 I2 + 10 I1 – 3 I3

Exercise 1.4.19 Solve the system of equations 9x-2y+4z=-17, 13x-3y+6z=-25, and -2x-z=3.

Water is flowing through a network of pipes (in thousands of cubic meters per hour), as shown in the figure. (Assume F₁ = 900 and F₂ = 800. If the system has an infinite number of solutions, express X1, X2, X3, X4, X5, X6, and x7 in terms of the parameters s and t.) (X1, X2, X3, X4, X5, X6, X7) = X4 (a) Solve this system for the water flow represented by x₁, i = 1, 2, ..., 7. - (C (b) Find the water flow when (X1, X2, X3, X4, X5, X6, X7) = X5 A x₁ = x2 = 200. (c) Find the water flow when x6 x7 = 0. (X1, X2, X3, X4, X5, X6, X7) = (d) Find the water flow when x5 = 1000 and x6 = 0. (X1, X2, X3, X4, X5, X6, X7) = |

Solve the systems of linear equations over the indicated Zp 3x + 2y = 1 over Z7  x + 4y = 1

Use the addition method to solve the following system of linear equa-tions:2x-3y = 143x + 2y = 8