Question

Using appropriate software, solve these systems of algebraic equations. (a) $$ \begin{aligned} 3 x_{1}+2 x_{2}-x_{3}+x_{4} &=6 \\ x_{1}+2 x_{2}-x_{4} &=-3 \\ -2 x_{1}+x_{2}+3 x_{3}+x_{4} &=2 \\ 3 x_{2}+x_{3}-4 x_{4} &=-6 \end{aligned} $$ (b) $$ \begin{aligned} 3 x_{1}+x_{2}^{2}+2 x_{3} &=8 \\ -x_{1}^{2}+3 x_{2}+2 x_{3} &=-6.293 \\ 2 x_{1}-x_{2}^{4}+4 x_{3} &=-12 \end{aligned} $$

Short answer

To solve the given systems: 1. For the first system of linear equations (a), rewrite it in matrix form and use software like MATLAB, Python (with NumPy library), or Wolfram Alpha to find the solution. 2. For the second system of nonlinear equations (b), rewrite it as a system of functions and use software like Mathematica, Python (with scipy library), or Wolfram Alpha to find the numerical solution. Keep in mind that the solutions for the nonlinear system (b) may depend on the initial guess used.

Step-by-step solution

Setup the first system of linear equations (a)

Re-write the system (a) in matrix form: $$ \begin{pmatrix} 3 & 2 & -1 & 1 \\ 1 & 2 & 0 & -1 \\ -2 & 1 & 3 & 1 \\ 0 & 3 & 1 & -4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \\ 2 \\ -6 \end{pmatrix} $$

Solve the first system of linear equations (a) using appropriate software

You may use MATLAB, Python (with NumPy library), or Wolfram Alpha to solve this system of linear equations. For example, in Python with NumPy library: ```python import numpy as np A = np.array([[3, 2, -1, 1], [1, 2, 0, -1], [-2, 1, 3, 1], [0, 3, 1, -4]]) b = np.array([6, -3, 2, -6]) solution = np.linalg.solve(A, b) print(solution) ```

Setup the second system of nonlinear equations (b)

Re-write the system (b) as a system of functions: $$ \begin{cases} f_1(x_1, x_2, x_3) = 3x_1 + x_2^2 + 2x_3 - 8 \\ f_2(x_1, x_2, x_3) = -x_1^2 + 3x_2 + 2x_3 + 6.293 \\ f_3(x_1, x_2, x_3) = 2x_1 - x_2^4 + 4x_3 + 12 \end{cases} $$

Solve the second system of nonlinear equations (b) using appropriate software

You may use Mathematica, Python (with scipy library), or Wolfram Alpha to find the numerical solution of this system of nonlinear equations. For example, in Python with scipy library: ```python from scipy.optimize import fsolve def system(x): x1, x2, x3 = x f1 = 3*x1 + x2**2 + 2*x3 - 8 f2 = -x1**2 + 3*x2 + 2*x3 + 6.293 f3 = 2*x1 - x2**4 + 4*x3 + 12 return [f1, f2, f3] initial_guess = (1, 1, 1) solution = fsolve(system, initial_guess) print(solution) ``` Remember that the solutions obtained may depend on the initial guess in the case of nonlinear systems.