To solve the system of linear equations:

1. $x_1 + x_2 - x_3 + 3x_4 - x_5 = -14$
2. $2x_1 + x_2 + x_3 + x_4 + x_5 = 16$
3. $x_1 + x_2 - x_3 + 2x_4 - x_5 = -10$
4. $2x_1 + x_2 + 4x_3 + x_4 - x_5 = 18$
5. $3x_1 + x_2 + x_3 - 2x_4 + x_5 = 32$

I will solve this system using matrix methods. numpy as np

Coefficient matrix (A)

A = np.array([ [1, 1, -1, 3, -1], [2, 1, 1, 1, 1], [1, 1, -1, 2, -1], [2, 1, 4, 1, -1], [3, 1, 1, -2, 1] ])

Constants vector (B)

B = np.array([-14, 16, -10, 18, 32])

Solving the system

solution = np.linalg.solve(A, B) solutionThe solution to the system of equations is:

$x_1 = 4, \quad x_2 = 3, \quad x_3 = 4, \quad x_4 = -4, \quad x_5 = 5$

Would you like detailed steps for how this solution was found, or do you have additional questions?

Here are 5 related questions to deepen your understanding:

1. How can you verify the solution to a system of equations?
2. What are the different methods to solve a system of linear equations (e.g., substitution, elimination, matrices)?
3. How does the determinant of a matrix affect the solvability of a system of equations?
4. What happens if the system of equations has no solution or infinite solutions?
5. How can you solve a larger system of linear equations efficiently?

Tip: To verify your solution, substitute the values back into the original equations and check if they satisfy all equations.