Simultaneous Linear Equations/Examples/Arbitrary System 10

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Example of Simultaneous Linear Equations

Let $S$ denote the system of simultaneous linear equations:

\(\ds x + 2 y - z + w\) \(=\) \(\ds 1\)
\(\ds 2 x + y + z\) \(=\) \(\ds 2\)


$S$ has as its solution set:

\(\ds 3 x\) \(=\) \(\ds 3 - z + w\)
\(\ds 3 y\) \(=\) \(\ds 3 z - 2 w\)

where $z$ and $w$ can be any numbers.


Proof

We express $S$ in matrix representation:

$\begin {pmatrix} 1 & 2 & -1 & 1 \\ 2 & 1 & 1 & 0 \end {pmatrix} \begin {pmatrix} x \\ y \\ z \\ w \end {pmatrix} = \begin {pmatrix} 1 \\ 2 \end {pmatrix}$

and consider the augmented matrix:

$\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix} = \paren {\begin {array} {cccc|c} 1 & 2 & -1 & 1 & 1 \\ 2 & 1 & 1 & 0 & 2 \end {array} }$


In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementary row operations that are to be applied to $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$.

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\begin {pmatrix} \mathbf A_k & \mathbf b_k \end {pmatrix}$.


$e_1 := r_2 \to r_2 - 2 r_1$

Hence:

$\begin {pmatrix} \mathbf A_1 & \mathbf b_1 \end {pmatrix} = \paren {\begin {array} {cccc|c}

1 & 2 & -1 & 1 & 1 \\ 0 & -3 & 3 & -2 & 0 \\ \end {array} }$


$e_2 := r_2 \to -\dfrac 1 3 r_2$

Hence:

$\begin {pmatrix} \mathbf A_2 & \mathbf b_2 \end {pmatrix} = \paren {\begin {array} {cccc|c}

1 & 2 & -1 & 1 & 1 \\ 0 & 1 & -1 & 2/3 & 0 \\ \end {array} }$


$e_3 := r_1 \to r_1 - 2 r_2$

Hence:

$\begin {pmatrix} \mathbf A_3 & \mathbf b_3 \end {pmatrix} = \paren {\begin {array} {cccc|c}

1 & 0 & 1 & -1/3 & 1 \\ 0 & 1 & -1 & 2/3 & 0 \\ \end {array} }$


$e_4 := r_1 \to 3 r_1$

$e_5 := r_2 \to 3 r_2$

Hence:

$\begin {pmatrix} \mathbf A_5 & \mathbf b_5 \end {pmatrix} = \paren {\begin {array} {cccc|c}

3 & 0 & 1 & -1 & 3 \\ 0 & 3 & -3 & 2 & 0 \\ \end {array} }$


Thus:

\(\ds 3 x + z - w\) \(=\) \(\ds 3\)
\(\ds 3 y - 3 z + 2 w\) \(=\) \(\ds 0\)

and it cannot be simplified further.

$\blacksquare$


Sources

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