In Exercises 37–44, use Cramer's Rule to solve each system. {4x−5y−6z=−1x−2y−5z=−122x−y=7\begin{cases} 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \end{cases}⎩⎨⎧4x−5y−6z=−1x−2y−5z=−122x−y=7

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 39

7장, 문제 39

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}\) 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \(\end{cases}$

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Write the system of equations in standard form, ensuring all variables are on the left side and constants on the right side: $\begin{cases} 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y + 0z = 7 \end{cases}$

Form the coefficient matrix $A$ from the coefficients of $x$, $y$, and $z$ in the system: $A = \begin{bmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{bmatrix}$

Calculate the determinant of matrix $A$, denoted as $\det(A)$, which is necessary to apply Cramer's Rule. This involves expanding the determinant using minors and cofactors.

Form matrices $A_x$, $A_y$, and $A_z$ by replacing the respective columns of $A$ with the constants vector $\mathbf{b} = \begin{bmatrix} -1 \\ -12 \\ 7 \end{bmatrix}$: - $A_x$ is formed by replacing the first column of $A$ with $\mathbf{b}$. - $A_y$ is formed by replacing the second column of $A$ with $\mathbf{b}$. - $A_z$ is formed by replacing the third column of $A$ with $\mathbf{b}$.

Calculate the determinants $\det(A_x)$, $\det(A_y)$, and $\det(A_z)$. Then, use Cramer's Rule to find the solutions: $x = \frac{\det(A_x)}{\det(A)}$, $y = \frac{\det(A_y)}{\det(A)}$, $z = \frac{\det(A_z)}{\det(A)}$.

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주요 개념

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

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the coefficient matrix has a non-zero determinant. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.

Determinants of Matrices

The determinant is a scalar value that can be computed from a square matrix and provides important properties about the matrix, such as invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula involving minors and cofactors. A non-zero determinant indicates the system has a unique solution.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, matrix operations, and Cramer's Rule, especially useful for small systems.

In Exercises 37–44, use Cramer's Rule to solve each system. {4x−5y−6z=−1x−2y−5z=−122x−y=7\(\begin{cases}\) 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \(\end{cases}\)⎩⎨⎧4x−5y−6z=−1x−2y−5z=−122x−y=7