Simultaneous linear equations with more than two variables
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Solve \(\begin{array}{*{20}{c}}{2x - 4y + z = 10}\\{x + 2y - z = 1}\\{ - x - 3y + 2z = 0}\end{array}\)
$$\begin{align}
&\text{Set up the Matrix from the coefficients}\\
&\left[\begin{array}{cccc}
2 & -4 & 1 & 10 \\
1 & 2 & -1 & 1 \\
-1 & -3 & 2 & 0
\end{array}\right]\\
&\text{Now using Gaussian elimination method:}\\
&\text{Add row 3 to row 2}\\
&\left[\begin{array}{cccc}
2 & -4 & 1 & 10 \\
0 & -1 & 1 & 1 \\
-1 & -3 & 2 & 10
\end{array}\right]\\
&\text{Add row 1 to}\,2\times\text{row} 3\\
&\left[\begin{array}{cccc}
2 & -4 & 1 & 0 \\
0 & -1 & 1 & 1 \\
0 & -10 & 5 & 10
\end{array}\right]\\
&\text{Divide row 3 by 5}\\
&\rightarrow\left[\begin{array}{llll}
2 & -4 & 1 & 10 \\
0 & -1 & 1 & 1 \\
0 & -2 & 1 & 2
\end{array}\right]\\
&\text { Subtract}\ 2\times\text{row } 2 \text { from row3 }\\
&\left[\begin{array}{rrrr}
2 & -4 & 1 & 10 \\
0 & -1 & 1 & 1 \\
0 & 0 & -1 & 0
\end{array}\right]\\
&\text{Divide row 1 by 2}\\
&\left[\begin{array}{ccc}1 & -2 & \dfrac{1}{2} & 5 \\
0 & -1 & 1 & 1 \\ 0 & 0 & -1 & 0\end{array}\right]\\
&\therefore\ z=0,-y+z=1,\ y=-1\\
&x-2 y+\frac{1}{2} z=5,\ x+2+0=5,\ x=3\\
&\therefore\ x = 3,\ y =-1,\ z=0
\end{align}
$$
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