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Obtain an equivalent system by performing the stated elementary operation on the system. Replace the second equation by the sum of itself and 3 times the first equation. \[ \begin{array}{l} 9 x+2 y-8 z=-36 \\ 3 x+5 y+6 z=-6 \\ 4 x+11 y-3 z=-20 \end{array} \] \[ \begin{array}{l} 9 x+2 y-8 z=-36 \quad \square x+\square y-\square z=\square \\ 3 x+5 y+6 z=-6 \rightarrow \square x+\square y-\square z=\square \\ 4 x+11 y-3 z=-20 \end{array} \quad \square x+\square y-\square z=\square \square \] (Type integers or simplified fractions.)

Ask by Mullins Blake. in the United States
Feb 18,2025

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\[ \begin{array}{l} 9x + 2y - 8z = -36 \\ 30x + 11y - 18z = -114 \\ 4x + 11y - 3z = -20 \end{array} \]

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To obtain the equivalent system, we need to replace the second equation by the sum of itself and 3 times the first equation. First, we calculate 3 times the first equation: \[ 3 \cdot (9 x + 2 y - 8 z = -36) \Rightarrow 27 x + 6 y - 24 z = -108. \] Now we add this to the second equation: \[ (3 x + 5 y + 6 z) + (27 x + 6 y - 24 z) = -6 - 108. \] Combining the parts gives: \[ (3 x + 27 x) + (5 y + 6 y) + (6 z - 24 z) = -6 - 108, \] which simplifies to: \[ 30 x + 11 y - 18 z = -114. \] So, the equivalent system is: \[ \begin{array}{l} 9 x + 2 y - 8 z = -36 \quad \\ 30 x + 11 y - 18 z = -114 \quad \\ 4 x + 11 y - 3 z = -20 \end{array} \]

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