In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. x + 2y + 3z = 5 y - 5z = 0

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1. Step 1: Write the system of equations in augmented matrix form: \( \begin{bmatrix} 1 & 2 & 3 & | & 5 \\ 0 & 1 & -5 & | & 0 \end{bmatrix} \).
2. Step 2: Use the second row to eliminate the \( y \) term in the first row. Since the second row already has a leading 1 in the \( y \) position, subtract 2 times the second row from the first row.
3. Step 3: Update the first row after the elimination: \( \begin{bmatrix} 1 & 0 & 13 & | & 5 \\ 0 & 1 & -5 & | & 0 \end{bmatrix} \).
4. Step 4: Express the system of equations from the row-reduced form: \( x + 13z = 5 \) and \( y - 5z = 0 \).
5. Step 5: Solve for \( x \) and \( y \) in terms of \( z \), where \( z \) is a free variable: \( x = 5 - 13z \) and \( y = 5z \).

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 applying back substitution to find the solutions of the system of equations.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is the set of values for the variables that satisfy all equations simultaneously. Systems can have one unique solution, infinitely many solutions, or no solution, depending on the relationships between the equations.