To write the given system of equations in matrix form, we need to express it as \( AX = B \), where \( A \) is the coefficients matrix, \( X \) is the variable matrix, and \( B \) is the constants matrix.
The given system of equations is:
- \( 2x + y + 0z = 30 \)
- \( x + 0y + 4z = 20 \)
- \( 0x + y + 2z = 10 \)
From this, we can identify the coefficients and constant terms:
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The coefficients matrix \( A \) is: \[ \begin{bmatrix} 2 & 1 & 0 \ 1 & 0 & 4 \ 0 & 1 & 2 \end{bmatrix} \]
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The variable matrix \( X \) is: \[ \begin{bmatrix} x \ y \ z \end{bmatrix} \]
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The constants matrix \( B \) is: \[ \begin{bmatrix} 30 \ 20 \ 10 \end{bmatrix} \]
Therefore, the augmented matrix representation of the system of equations will be:
\[ \begin{bmatrix} 2 & 1 & 0 & | & 30 \ 1 & 0 & 4 & | & 20 \ 0 & 1 & 2 & | & 10 \end{bmatrix} \]
From the responses you provided, the correct representation in the form of a 3 by 4 matrix is: \[ \begin{bmatrix} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{bmatrix} \]
So the correct option is: \[ \begin{bmatrix} 2 & 1 & 0 & 30 \ 1 & 0 & 4 & 20 \ 0 & 1 & 2 & 10 \end{bmatrix} \] which corresponds to the first matrix option provided in the question.