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$Solve the System Equations Using Gaussian {:[x_(1)+2x_(2)+6x_(3)=1],[2x_(2)+5x_(2)+15x_(3)=4],[3x_(1)+x_(2)+3x_(3)=-6]:}$

Solve the System Equations Using Gaussian $\newline$ $\begin{array}{l} x_{1}+2 x_{2}+6 x_{3}=1 \\ 2 x_{2}+5 x_{2}+15 x_{3}=4 \\ 3 x_{1}+x_{2}+3 x_{3}=-6 \end{array}$

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Q. Solve the System Equations Using Gaussian

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\begin{array}{l} x_{1}+2 x_{2}+6 x_{3}=1 \\ 2 x_{2}+5 x_{2}+15 x_{3}=4 \\ 3 x_{1}+x_{2}+3 x_{3}=-6 \end{array}

Write Augmented Matrix: Write down the augmented matrix for the system of equations. $\newline$ $\begin{bmatrix} 1 & 2 & 6 & | & 1 \\ 0 & 2 & 5 & | & 4 \\ 3 & 1 & 3 & | & -6 \end{bmatrix}$
Leading $1$ in First Row: Perform row operations to get a leading $1$ in the first row, first column. This is already done since the element is already $1$ .
Eliminate x_1 Terms: Use the leading $1$ in the first row to eliminate the x_1 terms from the other rows. Multiply the first row by $-3$ and add it to the third row. $\newline$ $R3 = R3 + (-3) * R1$ $\newline$ $\begin{bmatrix} 1 & 2 & 6 & | & 1 \\ 0 & 2 & 5 & | & 4 \\ 0 & -5 & -15 & | & -9 \end{bmatrix}$
Correct Third Row: There's a mistake in the previous step, the third row should be: $\newline$ $R3 = R3 + (-3) * R1$ $\newline$ $\begin{bmatrix} 1 & 2 & 6 & | & 1 \\ 0 & 2 & 5 & | & 4 \\ 0 & -5 & -15 & | & -9 \end{bmatrix}$ $\newline$ But the correct calculation is: $\newline$ $R3 = R3 + (-3) * R1 = (3 - 3*1, 1 - 3*2, 3 - 3*6, -6 - 3*1)$ $\newline$ $R3 = (0, -5, -15, -9)$ $\newline$ This is incorrect; the correct third row should be: $\newline$ $R3 = (0, -5, -15, -9)$ $\newline$ $\begin{bmatrix} 1 & 2 & 6 & | & 1 \\ 0 & 2 & 5 & | & 4 \\ 0 & -5 & -15 & | & -9 \end{bmatrix}$