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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 6 - Matrices and Determinants문제 55
7장, 문제 55

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

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Step 1: Understand that the determinant D corresponds to the coefficient matrix of the system of linear equations. The matrix for D is given as \( \begin{bmatrix} 2 & -4 \\ 3 & 5 \end{bmatrix} \). This means the system has two variables, say \( x \) and \( y \), and the coefficients of these variables in the two equations are from this matrix.
Step 2: The determinant \( D_x \) is formed by replacing the first column of the coefficient matrix with the constants from the right-hand side of the equations. The matrix for \( D_x \) is \( \begin{bmatrix} 8 & -4 \\ -10 & 5 \end{bmatrix} \). This tells us the constants on the right side of the equations are 8 and -10 respectively.
Step 3: Write the system of linear equations using the coefficients from matrix D and the constants from the right side. The first equation uses the first row of D: \( 2x - 4y = 8 \). The second equation uses the second row of D: \( 3x + 5y = -10 \).
Step 4: Verify that the system is consistent with the determinants given. The determinant D should not be zero for Cramer's Rule to apply, and the determinants D and \( D_x \) correspond to the coefficient matrix and the matrix with the first column replaced by constants, respectively.
Step 5: Summarize the system of equations as: \( \begin{cases} 2x - 4y = 8 \\ 3x + 5y = -10 \end{cases} \). This is the system for which Cramer's Rule yields the given determinants.

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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 states that each variable in the system can be found by dividing the determinant of a matrix formed by replacing the variable's column with the constants column by the determinant of the coefficient matrix.
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6:54
Cramer's Rule - 2 Equations with 2 Unknowns

Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine if the system has a unique solution (non-zero determinant) and is essential in applying Cramer's Rule.
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가이드 코스
4:36
Determinants of 2×2 Matrices

Forming Systems of Linear Equations from Determinants

Given determinants D and D_x, you can reconstruct the system of linear equations by identifying the coefficient matrix from D and replacing the appropriate column with constants from D_x. This process helps write the original system that corresponds to the given determinants.
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가이드 코스
4:27
Introduction to Systems of Linear Equations
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교과서 질문

Evaluate each determinant.

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교과서 질문

Use Cramer's Rule to solve each system.

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교과서 질문

In Exercises 57–60, solve each equation for x.

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교과서 질문

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교과서 질문

In Exercises 57–60, solve each equation for x.

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교과서 질문

Solve the system

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