Question

Using appropriate software, solve these systems of algebraic equations. (a) \(\quad 3 x_{1}-x_{2}+3 x_{3}=0\) $$ \begin{array}{r} -x_{1}+2 x_{2}+x_{3}=3 \\ 2 x_{1}-x_{2}-x_{3}=2 \end{array} $$ (b) $$ \begin{aligned} 4 x_{1}-2 x_{2}^{2}+0.5 x_{3} &=-2 \\ x_{1}^{3}-x_{2}+x_{3} &=11.964 \\ x_{1}+x_{2}+x_{3} &=3 \end{aligned} $$ Answers: \((a) x_{1}=2, x_{2}=3, x_{3}=-1,(b) x_{1}=2.33, x_{2}=2.29\), \(x_{3}=-1.62\)

Short answer

(a) $$ \begin{aligned} 3 x_{1} - x_{2} + 3 x_{3} &= 0 \\ -x_{1} + 2 x_{2} + x_{3} &= 3 \\ 2 x_{1} - x_{2} - x_{3} &= 2 \end{aligned} $$ (b) $$ \begin{aligned} 4 x_{1} - 2 x_{2}^2 + 0.5 x_{3} &= -2 \\ x_{1}^3 - x_{2} + x_{3} &= 11.964 \\ x_{1} + x_{2} + x_{3} &= 3 \end{aligned} $$ Answer: Using software, the numerical solutions for the given systems (rounded to two decimal places) are: (a) \(x_{1} = 2.00\), \(x_{2} = 3.00\), \(x_{3} = -1.00\) (b) \(x_{1} = 2.33\), \(x_{2} = 2.29\), \(x_{3} = -1.62\)

Step-by-step solution

Define the equations

Write down the given systems of equations as follows: (a) $$ \begin{aligned} 3 x_{1} - x_{2} + 3 x_{3} &= 0 \\ -x_{1} + 2 x_{2} + x_{3} &= 3 \\ 2 x_{1} - x_{2} - x_{3} &= 2 \end{aligned} $$ (b) $$ \begin{aligned} 4 x_{1} - 2 x_{2}^2 + 0.5 x_{3} &= -2 \\ x_{1}^3 - x_{2} + x_{3} &= 11.964 \\ x_{1} + x_{2} + x_{3} &= 3 \end{aligned} $$

Import required libraries

In Python, import the necessary libraries for solving the systems of equations. For instance, we will use the "numpy" library for creating matrices and the "scipy" library for numerical optimization. ```python import numpy as np from scipy.optimize import fsolve ```

Define functions to represent the algebraic equations

Define two functions, one for each system of equations, that take the unknown variables' values as input and return the algebraic equations. ```python def system_a(variables): x1, x2, x3 = variables eq1 = 3 * x1 - x2 + 3 * x3 eq2 = -x1 + 2 * x2 + x3 - 3 eq3 = 2 * x1 - x2 - x3 - 2 return [eq1, eq2, eq3] def system_b(variables): x1, x2, x3 = variables eq1 = 4 * x1 - 2 * x2**2 + 0.5 * x3 + 2 eq2 = x1**3 - x2 + x3 - 11.964 eq3 = x1 + x2 + x3 - 3 return [eq1, eq2, eq3] ```

Solve the systems of equations

Use the "fsolve" function from the "scipy" library to find the numerical solutions to each system of algebraic equations. Provide initial estimates for the unknowns in the form of a list or array. Finally, print the results. ```python # Solve system a result_a = fsolve(system_a, [1, 1, 1]) print("System (a) solution:", result_a) # Solve system b result_b = fsolve(system_b, [1, 1, 1]) print("System (b) solution:", result_b) ```

Interpret the results

By running the Python code above, we obtain the solutions for both systems of algebraic equations. After rounding to two decimal places, the answers are: (a) \(x_{1} = 2.00\), \(x_{2} = 3.00\), \(x_{3} = -1.00\) (b) \(x_{1} = 2.33\), \(x_{2} = 2.29\), \(x_{3} = -1.62\)