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

검증된 단계별 안내

Step 1: Write the system of equations in augmented matrix form. This involves organizing the coefficients of the variables and the constants into a matrix. For example, if the system is: \(a_1x + b_1y + c_1z = d_1\), \(a_2x + b_2y + c_2z = d_2\), \(a_3x + b_3y + c_3z = d_3\), the augmented matrix would be: \(\begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{bmatrix}\).

Step 2: Perform row operations to achieve row echelon form. Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of rows. The goal is to create zeros below the pivot positions (leading coefficients) in each column.

Step 3: Continue row operations to achieve reduced row echelon form. This involves ensuring that each pivot is 1 and that all entries above and below each pivot are zero. Reduced row echelon form makes it easier to interpret the solution.

Step 4: Analyze the resulting matrix. If a row corresponds to an equation like \(0 = k\) (where \(k\) is a nonzero constant), the system has no solution. If the matrix has fewer pivot positions than variables, the system has infinitely many solutions, and you can express the solution in terms of free variables.

Step 5: Write the solution based on the reduced row echelon form. If the system has a unique solution, express the values of the variables directly. If there are infinitely many solutions, express the dependent variables in terms of the free variables.

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주요 개념

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Gaussian Elimination

Gaussian elimination is a systematic 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 equations, 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 form is crucial in Gaussian elimination as it allows for back substitution to find the values of the variables in a system of equations.

Existence of Solutions

The existence of solutions in a system of linear equations can be determined through the rank of the coefficient matrix and the augmented matrix. If the rank of both matrices is equal and less than or equal to the number of variables, the system has at least one solution. Conversely, if the rank of the augmented matrix exceeds that of the coefficient matrix, the system is inconsistent and has no solutions.