Question

For the following linear system, put the augmented coefficient matrix into row-echelon form, and then use back substitution to find all solutions. (If the system is inconsistent, enter INCONSISTENT. If there are an infinite number of solutions use t as your parameter.) 2X1 + 8x2 + 3x3 = 2 X1 + 3x2 + 2x3 2X1 + 7x2 + 4x3 = 5 = 8 (X1, X2, X3) =

Answer 1

To solve the given linear system, one must write the augmented coefficient matrix, convert it to row-echelon form using elementary row operations, and then use back substitution to find the solutions, which could be a unique solution, no solution (inconsistent), or infinitely many solutions using a parameter 't'.

Step-by-step explanation:

To solve the given system of linear equations:

2X1 + 8X2 + 3X3 = 2

X1 + 3X2 + 2X3 = 8

2X1 + 7X2 + 4X3 = 5,

we first write the augmented coefficient matrix and then transform it into row-echelon form. We use operations such as row swapping, scaling rows, and adding multiples of one row to another. Once in row-echelon form, back substitution allows us to find the solutions for X3, X2, and X1 in that order.

Step-by-Step Process

1. Write the augmented matrix for the system of equations.
2. Use elementary row operations to reach row-echelon form.
3. Perform back substitution to find the solutions to the system.

We might find that the system has a unique solution, no solution (in which case it's inconsistent), or infinitely many solutions using a parameter t to represent the free variable. After reaching row-echelon form, it would be clear which case applies to this system.