Question

Using appropriate software, solve these systems of algebraic equations. (a) $$ \begin{aligned} 4 x_{1}-x_{2}+2 x_{3}+x_{4} &=-6 \\ x_{1}+3 x_{2}-x_{3}+4 x_{4} &=-1 \\ -x_{1}+2 x_{2}+5 x_{4} &=5 \\ 2 x_{2}-4 x_{3}-3 x_{4} &=-5 \end{aligned} $$ (b) $$ \begin{aligned} 2 x_{1}+x_{2}^{4}-2 x_{3}+x_{4} &=1 \\ x_{1}^{2}+4 x_{2}+2 x_{3}^{2}-2 x_{4} &=-3 \\ -x_{1}+x_{2}^{4}+5 x_{3} &=10 \\ 3 x_{1}-x_{3}^{2}+8 x_{4} &=15 \end{aligned} $$

Short answer

Answer: The key differences in solving a system of linear and nonlinear equations using a computer software are the methods that the software uses, and the type and number of solutions that may be obtained. For linear systems, the solutions are unique and can be found through methods like matrix inversion. For nonlinear systems, the software may provide multiple solutions or numerical approximations, and the methods used can involve techniques like iterative solvers or symbolic manipulation. It is important to understand the software's limitations and syntax when solving both types of systems.

Step-by-step solution

(a) Solving the linear system using software

To solve the linear system of algebraic equations, input the following system in a software of your choice (Wolfram|Alpha, MATLAB, etc.): $$ \begin{aligned} 4 x_{1}-x_{2}+2 x_{3}+x_{4} &=-6 \\\ x_{1}+3 x_{2}-x_{3}+4 x_{4} &=-1 \\\ -x_{1}+2 x_{2}+5 x_{4} &=5 \\\ 2 x_{2}-4 x_{3}-3 x_{4} &=-5 \end{aligned} $$ The software will provide the solutions for the variables \(x_1, x_2, x_3,\) and \(x_4\). Note: Each software may have different syntax for inputting the system of equations. Refer to the software documentation or help files for the correct syntax.

(b) Solving the nonlinear system using software

To solve the nonlinear system of algebraic equations, input the following system in a software of your choice (Wolfram|Alpha, MATLAB, etc.): $$ \begin{aligned} 2 x_{1}+x_{2}^{4}-2 x_{3}+x_{4} &=1 \\\ x_{1}^{2}+4 x_{2}+2 x_{3}^{2}-2 x_{4} &=-3 \\\ -x_{1}+x_{2}^{4}+5 x_{3} &=10 \\\ 3 x_{1}-x_{3}^{2}+8 x_{4} &=15 \end{aligned} $$ The software will provide the solutions for the variables \(x_1, x_2, x_3,\) and \(x_4\). Note: For nonlinear systems, the software may provide multiple solutions or numerical approximations depending on its capabilities and settings. Ensure that you understand the particular software's limitations and follow its syntax when inputting the system of equations. Remember to verify the solutions provided by the software by substituting them back into the original equations to check if the equations are satisfied.