Find the values of the variables for which each statement is true, if possible.[5x+2−6yz]=[a3x−15y9]\left[ \begin{matrix} 5 & x+2 \\ -6y & z \end{matrix} \right] = \left[ \begin{matrix} a & 3x-1 \\ 5y & 9 \end{matrix} \right]

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 71

6장, 문제 71

Find the values of the variables for which each statement is true, if possible.
$[\begin{matrix} 5 & x + 2 \\ - 6 y & z \end{matrix}] = [\begin{matrix} a & 3 x - 1 \\ 5 y & 9 \end{matrix}]$

검증된 단계별 안내

Identify the corresponding elements in each matrix. For two matrices to be equal, their elements in the same positions must be equal. For example, if the matrices are $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ and $ \begin{bmatrix} e & f \\ g & h \end{bmatrix} $, then you set up the equations $ a = e $, $ b = f $, $ c = g $, and $ d = h $.

Write down the system of equations by equating each corresponding element from the two matrices. This will give you multiple equations involving the variables.

Solve the system of equations step-by-step. You can use substitution or elimination methods to find the values of the variables that satisfy all equations simultaneously.

Check for consistency in the system. If the system has no solution, then the matrices cannot be equal for any values of the variables. If there is a solution, those values make the matrices equal.

Verify your solution by substituting the found values back into the original matrices to ensure that all corresponding elements are indeed equal.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Matrix Equality

Two matrices are equal if and only if they have the same dimensions and their corresponding entries are equal. For 2x2 matrices, this means each element in the first matrix must be equal to the corresponding element in the second matrix.

Solving Systems of Equations

When matrix entries contain variables, equating corresponding elements forms a system of equations. Solving this system involves finding values of variables that satisfy all equations simultaneously.

Substitution and Simplification

To solve for variables, substitution or algebraic manipulation is used to simplify equations. This process helps isolate variables and find consistent solutions or determine if no solution exists.

Find the values of the variables for which each statement is true, if possible.[5x+2−6yz]=[a3x−15y9]\(\left\)[ \(\begin{matrix}\) 5 & x+2 \\ -6y & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) a & 3x-1 \\ 5y & 9 \(\end{matrix}\) \(\right\)]