Solve a system of four linear equations in four variables

Published on December 7, 2024

Linear-Algebra

The solution to the system of linear equations is x = 1, y = -2, z = 3, and w = 2.

Question

Answer

Here’s how to solve the system of linear equations:

The system of equations is:

Subtracting equation (4) from equation (3):

Substitute

into equation (4):

(5)

Substitute

into equation (1):

(6)

Subtracting equation (6) from equation (5):

Substitute

into equation (6):

(7)

Substitute

and

into equation (2):

(8)

Multiply equation (7) by 2:

(9)

Subtract equation (9) from equation (8):

Substitute

into equation (7):

So, the solution is

Since the system has a unique solution, it is a consistent and independent system.

Final Answer: The final answer is

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