Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.[1141∣3−11−10∣72005∣1100124∣5]\begin{bmatrix} 1 & 1 & 4 & 1 & \vert & 3 \\ -1 & 1 & -1 & 0 & \vert & 7 \\ 2 & 0 & 0 & 5 & \vert & 11 \\ 0 & 0 & 12 & 4 & \vert & 5 \end{bmatrix}1−12011004−10121054∣∣∣∣37115

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 11

7장, 문제 11

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.
$\begin{bmatrix}\) 1 & 1 & 4 & 1 & $\vert$ & 3 \\ -1 & 1 & -1 & 0 & $\vert$ & 7 \\ 2 & 0 & 0 & 5 & $\vert$ & 11 \\ 0 & 0 & 12 & 4 & $\vert$ & 5 \(\end{bmatrix}$

검증된 단계별 안내

Identify the variables corresponding to each column in the augmented matrix. Since there are four columns before the augmented part, use variables w, x, y, and z in that order.

Write the first row as an equation by multiplying each variable by the corresponding coefficient and setting it equal to the augmented value: $1w + 1x + 4y + 1z = 3$.

Write the second row as an equation: $-1w + 1x - 1y + 0z = 7$.

Write the third row as an equation: $0w + 2x + 0y + 5z = 11$.

Write the fourth row as an equation: $0w + 0x + 12y + 4z = 5$.

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주요 개념

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

Augmented Matrix Representation

An augmented matrix combines the coefficients of variables and constants from a system of linear equations into a single matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants. This format simplifies solving systems using matrix operations.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Writing the system from an augmented matrix involves translating each row into an equation.

Variables and Their Correspondence

Variables in the system correspond to the columns of the augmented matrix, excluding the last column which represents constants. In this problem, the variables are w, x, y, and z, matching the four columns of coefficients. Correctly identifying variables is essential to write the system accurately.