In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 5x + 12y + z = 10 2x + 5y + 2z = - 1 x + 2y - 3z = 5

Verified step by step guidance

1. Step 1: Write the system of equations as an augmented matrix.
2. Step 2: Use row operations to get a leading 1 in the first row, first column.
3. Step 3: Eliminate the first column entries below the leading 1 by using row operations.
4. Step 4: Use row operations to get a leading 1 in the second row, second column, and eliminate the second column entries below this leading 1.
5. Step 5: Use row operations to get a leading 1 in the third row, third column, and eliminate the third column entries above this leading 1 to achieve row-echelon form.

Key Concepts

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It involves transforming the system's augmented matrix into row echelon form using a series of row operations, which include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. This process simplifies the system, making it easier to find solutions or determine if no solution exists.

Row Echelon Form

Row echelon form is a specific arrangement of a matrix where all non-zero rows are above any rows of all zeros, and the leading coefficient of each non-zero row (the first non-zero number from the left) is to the right of the leading coefficient of the previous row. This structure is crucial for identifying the rank of the matrix and determining the number of solutions to the system of equations.

Consistency of a System

A system of equations is considered consistent if it has at least one solution, while it is inconsistent if no solutions exist. The consistency can be determined through the final form of the augmented matrix after applying Gaussian elimination. If a row leads to a contradiction, such as 0 = c (where c is a non-zero constant), the system is inconsistent; otherwise, it may have a unique or infinitely many solutions.