In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

검증된 단계별 안내

Step 1: Write the system of equations in matrix form as A * X = B, where A is the coefficient matrix, X is the variable vector, and B is the constants vector. Here, A = [[2, -3, 2], [2, 3, -2], [2, -9, 6]], X = [x, y, z]^T, and B = [4, 6, 2]^T.

Step 2: Calculate the determinant of the coefficient matrix A, denoted as det(A). If det(A) = 0, Cramer's Rule cannot be used because it requires a non-zero determinant to find a unique solution.

Step 3: Since det(A) = 0 (as you will find upon calculation), explain that the system either has infinitely many solutions or no solution, so Cramer's Rule is not applicable.

Step 4: Use Gaussian elimination to solve the system. Start by writing the augmented matrix [A | B]: [[2, -3, 2, 4], [2, 3, -2, 6], [2, -9, 6, 2]].

Step 5: Perform row operations to reduce the augmented matrix to row-echelon form, then use back substitution to find the values of x, y, and z or determine if the system is inconsistent or dependent.

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주요 개념

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

Cramer's Rule and Determinants

Cramer's Rule solves a system of linear equations using determinants, applicable only when the coefficient matrix has a nonzero determinant. If the determinant is zero, the system is either dependent or inconsistent, making Cramer's Rule unusable.

Gaussian Elimination

Gaussian elimination is a method to solve systems of linear equations by transforming the augmented matrix into row-echelon form using row operations. This process simplifies the system, allowing back-substitution to find the solution or determine if none exists.

Consistency and Dependence of Systems

A system is consistent if it has at least one solution and inconsistent if it has none. Dependence occurs when equations are multiples or linear combinations of others, leading to infinite solutions or no unique solution, which affects the applicability of solution methods.