In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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Write the given system of linear equations in augmented matrix form. The augmented matrix combines the coefficients of the variables and the constants from the equations into a single matrix.

Use row operations (row swapping, scaling a row by a nonzero constant, or adding/subtracting multiples of one row to another) to transform the augmented matrix into row-echelon form. The goal is to create zeros below the pivot positions (leading entries in each row).

Continue using row operations to further simplify the matrix into reduced row-echelon form, where each pivot is 1, and all other entries in the pivot's column are 0.

Interpret the reduced row-echelon form of the matrix to determine the solution. If the system has a unique solution, express it in terms of the variables. If there are infinitely many solutions, express the solution in terms of free variables. If the system is inconsistent, state that no solution exists.

Verify the solution by substituting it back into the original system of equations to ensure all equations are satisfied.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 identify 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, as it clearly indicates the relationships between the variables.

Consistency of a System

A system of linear equations is considered consistent if it has at least one solution, and inconsistent if it has no solutions. During Gaussian elimination, if a row reduces to a form that implies a contradiction (such as 0 = 1), the system is inconsistent. Understanding the consistency of a system is essential for determining whether a complete solution can be found or if the system is unsolvable.