교과서 질문
Graph the solution set of each system of inequalities.
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Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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6장, 문제 67
Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
검증된 단계별 안내1
Write the system of equations in standard form:
\(5x + 4y = 10\) and \(3x - 7y = 6\).
Identify the coefficient matrix and calculate the determinant \(D\):
\(D = \begin{vmatrix} 5 & 4 \\ 3 & -7 \end{vmatrix} = (5)(-7) - (4)(3)\).
Calculate the determinant \(D_x\) by replacing the first column of the coefficient matrix with the constants from the right side of the equations:
\(D_x = \begin{vmatrix} 10 & 4 \\ 6 & -7 \end{vmatrix} = (10)(-7) - (4)(6)\).
Calculate the determinant \(D_y\) by replacing the second column of the coefficient matrix with the constants:
\(D_y = \begin{vmatrix} 5 & 10 \\ 3 & 6 \end{vmatrix} = (5)(6) - (10)(3)\).
Use Cramer's rule to find the solutions for \(x\) and \(y\):
\(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\). If \(D = 0\), then use another method such as substitution or elimination to find the solution set.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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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. For a system of two equations with two variables, it involves calculating the determinant of the coefficient matrix (D) and determinants of matrices formed by replacing columns with constants. If D ≠ 0, the system has a unique solution found by dividing these determinants.
추천 영상:
6:54
Cramer's Rule - 2 Equations with 2 Unknowns
Determinant of a 2x2 Matrix
The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value helps determine if the system of equations has a unique solution (nonzero determinant) or if it is dependent or inconsistent (zero determinant). It is essential for applying Cramer's Rule.
추천 영상:
4:36Determinants of 2×2 Matrices
Alternative Methods for Solving Systems When D = 0
If the determinant D equals zero, Cramer's Rule cannot be used because the system may have infinitely many solutions or no solution. In such cases, methods like substitution, elimination, or matrix row reduction are used to analyze the system and find the solution set or determine inconsistency.
추천 영상:
04:03Choosing a Method to Solve Quadratics
관련 실천
교과서 질문
Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
4x + 3y = -7
2x + 3y = -11
618
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교과서 질문
Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 1.5
x + 3y = 5
2x + 4y = 3
613
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교과서 질문
Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.3 and lateral surface area 65 in.2.
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교과서 질문
Graph the solution set of each system of inequalities.
531
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교과서 질문
Find each product, if possible.
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