## E3 - Solving linear systems

#### Example 1 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} & & &+& 2 \, x_{3} &+& 3 \, x_{4} &=& 3 \\ & & x_{2} &-& 2 \, x_{3} &+& 3 \, x_{4} &=& -1 \\ & & 2 \, x_{2} &-& 4 \, x_{3} &+& 6 \, x_{4} &=& -2 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 0 & 2 & 3 & 3 \\ 0 & 1 & -2 & 3 & -1 \\ 0 & 2 & -4 & 6 & -2 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 2 & 3 & 3 \\ 0 & 1 & -2 & 3 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \, a - 3 \, b + 3 \\ 2 \, a - 3 \, b - 1 \\ a \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 2 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} -x_{1} & & & & &=& 3 \\-3 \, x_{1} &+& x_{2} &+& 2 \, x_{3} &=& 6 \\2 \, x_{1} & & & & &=& -6 \\-3 \, x_{1} &+& 2 \, x_{2} &+& 4 \, x_{3} &=& 3 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} -1 & 0 & 0 & 3 \\ -3 & 1 & 2 & 6 \\ 2 & 0 & 0 & -6 \\ -3 & 2 & 4 & 3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \\ -2 \, a - 3 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 3 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &-& 5 \, x_{2} &-& 7 \, x_{3} &=& 3 \\-x_{1} &-& 4 \, x_{2} &-& 2 \, x_{3} &=& 6 \\ &-& 5 \, x_{2} &-& 5 \, x_{3} &=& 5 \\x_{1} &+& 3 \, x_{2} &+& x_{3} &=& -5 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -5 & -7 & 3 \\ -1 & -4 & -2 & 6 \\ 0 & -5 & -5 & 5 \\ 1 & 3 & 1 & -5 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -2 & -2 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 2 \\ -a - 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 4 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} -x_{1} &-& 4 \, x_{2} &+& 4 \, x_{3} &=& -7 \\2 \, x_{1} &+& 8 \, x_{2} &+& x_{3} &=& -4 \\ & & & & 4 \, x_{3} &=& -8 \\-x_{1} &-& 4 \, x_{2} &+& 2 \, x_{3} &=& -3 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} -1 & -4 & 4 & -7 \\ 2 & 8 & 1 & -4 \\ 0 & 0 & 4 & -8 \\ -1 & -4 & 2 & -3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 4 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -4 \, a - 1 \\ a \\ -2 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 5 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ 0 \\ -1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ 2 \\ 1 \end{array}\right] = \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 2 & -1 & -2 \\ 0 & 1 & 0 & -2 \\ -1 & -4 & 2 & 2 \\ 0 & 0 & 1 & -4 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ -4 \end{array}\right] \right\}$$

#### Example 6 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &+& x_{2} & & &=& 2 \\-2 \, x_{1} &-& x_{2} & & &=& -1 \\3 \, x_{1} & & &+& x_{3} &=& 1 \\-x_{1} &-& 4 \, x_{2} &+& x_{3} &=& -6 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 1 & 0 & 2 \\ -2 & -1 & 0 & -1 \\ 3 & 0 & 1 & 1 \\ -1 & -4 & 1 & -6 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

The solution set is $$\left\{\right\}$$

#### Example 7 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} & & &-& x_{3} &=& -2 \\x_{1} &-& 2 \, x_{2} &+& 5 \, x_{3} &=& 8 \\-2 \, x_{1} &+& 4 \, x_{2} &-& 6 \, x_{3} &=& -8 \\ & & &-& x_{3} &=& -2 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 0 & 0 & -1 & -2 \\ 1 & -2 & 5 & 8 \\ -2 & 4 & -6 & -8 \\ 0 & 0 & -1 & -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & -2 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 2 \\ a \\ 2 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 8 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -4 \\ -5 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ 5 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} -6 \\ -7 \\ -2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -8 \\ -4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} -4 & 1 & 4 & -6 & -7 \\ -5 & 1 & 5 & -7 & -8 \\ -4 & 0 & 5 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 3 & 1 \\ 0 & 1 & 0 & -2 & -3 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \, a + 1 \\ 2 \, a - 3 \\ -2 \, a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 9 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 0 \\ -1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 6 \\ -6 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -6 \\ 3 \\ -6 \end{array}\right] = \left[\begin{array}{c} 4 \\ -2 \\ 4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 0 & -2 & 6 & -6 & 4 \\ -1 & 1 & -6 & 3 & -2 \\ -1 & -2 & 3 & -6 & 4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 3 & 0 & 0 \\ 0 & 1 & -3 & 3 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \, a \\ 3 \, a - 3 \, b - 2 \\ a \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 10 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} 3 \, x_{1} &-& 7 \, x_{2} &+& 3 \, x_{3} &+& 4 \, x_{4} &=& -5 \\3 \, x_{1} &-& 8 \, x_{2} &+& 4 \, x_{3} &+& 4 \, x_{4} &=& -5 \\2 \, x_{1} &-& 4 \, x_{2} &+& x_{3} &+& 3 \, x_{4} &=& -4 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 3 & -7 & 3 & 4 & -5 \\ 3 & -8 & 4 & 4 & -5 \\ 2 & -4 & 1 & 3 & -4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 & 2 \\ 0 & 0 & 1 & -1 & 2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 1 \\ a + 2 \\ a + 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 11 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &-& 2 \, x_{2} &+& x_{3} &-& 5 \, x_{4} &=& -5 \\-x_{1} &+& 2 \, x_{2} &+& 4 \, x_{3} &-& 5 \, x_{4} &=& -5 \\ & & & & 4 \, x_{3} &-& 8 \, x_{4} &=& -8 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & -2 & 1 & -5 & -5 \\ -1 & 2 & 4 & -5 & -5 \\ 0 & 0 & 4 & -8 & -8 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & -2 & 0 & -3 & -3 \\ 0 & 0 & 1 & -2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a + 3 \, b - 3 \\ a \\ 2 \, b - 2 \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 12 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} -2 \, x_{1} &+& x_{2} &-& 8 \, x_{3} &=& -2 \\x_{1} &-& x_{2} &+& 5 \, x_{3} &=& 1 \\ &-& 4 \, x_{2} &+& 8 \, x_{3} &=& 0 \\-x_{1} &+& 2 \, x_{2} &-& 7 \, x_{3} &=& -1 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} -2 & 1 & -8 & -2 \\ 1 & -1 & 5 & 1 \\ 0 & -4 & 8 & 0 \\ -1 & 2 & -7 & -1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 3 & 1 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \, a + 1 \\ 2 \, a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 13 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 0 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -3 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 4 \\ 3 \\ 5 \end{array}\right] = \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 1 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -3 & 1 & -1 \\ -1 & -2 & 4 & 1 \\ 0 & -3 & 3 & 0 \\ -1 & -3 & 5 & 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 1 \\ a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 14 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -2 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 2 \\ 4 \end{array}\right] = \left[\begin{array}{c} -1 \\ -2 \\ 2 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 1 & 0 & 4 & -1 \\ 0 & 1 & -2 & 2 & -2 \\ 2 & 0 & 4 & 4 & 2 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 2 & 2 & 1 \\ 0 & 1 & -2 & 2 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \, a - 2 \, b + 1 \\ 2 \, a - 2 \, b - 2 \\ a \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 15 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} & & &+& 5 \, x_{3} &+& 2 \, x_{4} &=& -7 \\ & & x_{2} &+& 3 \, x_{3} &+& 4 \, x_{4} &=& -7 \\ & & & & & & x_{4} &=& -2 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 0 & 5 & 2 & -7 \\ 0 & 1 & 3 & 4 & -7 \\ 0 & 0 & 0 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 5 & 0 & -3 \\ 0 & 1 & 3 & 0 & 1 \\ 0 & 0 & 0 & 1 & -2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -5 \, a - 3 \\ -3 \, a + 1 \\ a \\ -2 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 16 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &+& 4 \, x_{2} &-& x_{3} &=& 4 \\x_{1} &+& 4 \, x_{2} & & &=& 1 \\-x_{1} &-& 4 \, x_{2} & & &=& -1 \\2 \, x_{1} &+& 8 \, x_{2} & & &=& 2 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 4 & -1 & 4 \\ 1 & 4 & 0 & 1 \\ -1 & -4 & 0 & -1 \\ 2 & 8 & 0 & 2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 4 & 0 & 1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -4 \, a + 1 \\ a \\ -3 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 17 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ 4 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 5 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 3 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -3 \\ 1 \end{array}\right] = \left[\begin{array}{c} -3 \\ -8 \\ 3 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 1 & 0 & -1 & -3 \\ 4 & 5 & 3 & -3 & -8 \\ -3 & -5 & -5 & 1 & 3 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & -1 \\ 0 & 1 & 0 & 1 & -2 \\ 0 & 0 & 1 & 0 & 2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 1 \\ -a - 2 \\ 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 18 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -1 \\ 4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 1 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 8 \\ -3 \\ 4 \end{array}\right] = \left[\begin{array}{c} 4 \\ 6 \\ 1 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} -1 & -1 & -2 & 8 & 4 \\ 4 & 4 & 1 & -3 & 6 \\ -1 & -1 & -1 & 4 & 1 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -a + 2 \\ a \\ 1 \\ 1 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 19 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &-& 3 \, x_{2} &-& 5 \, x_{3} &=& 7 \\x_{1} &-& 3 \, x_{2} &-& 4 \, x_{3} &=& 6 \\x_{1} &-& 3 \, x_{2} &-& x_{3} &=& 3 \\-x_{1} &+& 3 \, x_{2} &+& x_{3} &=& -3 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -3 & -5 & 7 \\ 1 & -3 & -4 & 6 \\ 1 & -3 & -1 & 3 \\ -1 & 3 & 1 & -3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & -3 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 3 \, a + 2 \\ a \\ -1 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 20 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 0 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 1 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 1 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 2 \\ -4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 4 & -1 & -3 & 0 \\ 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & -2 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 4 & 0 & -2 & 2 \\ 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -4 \, a + 2 \, b + 2 \\ a \\ -b + 2 \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 21 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} & & 2 \, x_{2} &-& 4 \, x_{3} &+& x_{4} &=& -2 \\x_{1} &+& 2 \, x_{2} &-& 7 \, x_{3} &+& 3 \, x_{4} &=& 3 \\-x_{1} &+& x_{2} &+& x_{3} &-& 2 \, x_{4} &=& -7 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 0 & 2 & -4 & 1 & -2 \\ 1 & 2 & -7 & 3 & 3 \\ -1 & 1 & 1 & -2 & -7 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -3 & 0 & 1 \\ 0 & 1 & -2 & 0 & -2 \\ 0 & 0 & 0 & 1 & 2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 3 \, a + 1 \\ 2 \, a - 2 \\ a \\ 2 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 22 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -3 \\ -5 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 0 & -2 & 0 \\ 0 & 1 & -3 & 1 \\ 1 & 1 & -5 & 1 \\ 0 & -1 & 3 & -1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -2 & 0 \\ 0 & 1 & -3 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a \\ 3 \, a + 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 23 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} 5 \, x_{1} &+& 4 \, x_{2} &+& 3 \, x_{3} &=& 0 \\ & & x_{2} &+& 2 \, x_{3} &=& 0 \\5 \, x_{1} &+& 6 \, x_{2} &+& 7 \, x_{3} &=& 0 \\2 \, x_{1} &+& x_{2} & & &=& 0 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 5 & 4 & 3 & 0 \\ 0 & 1 & 2 & 0 \\ 5 & 6 & 7 & 0 \\ 2 & 1 & 0 & 0 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & -1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} a \\ -2 \, a \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 24 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -1 \\ -1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 1 \\ 3 \\ 6 \\ -2 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} -1 & 0 & 0 & 1 \\ -2 & -1 & 0 & 3 \\ -2 & -1 & 1 & 6 \\ 1 & 1 & 0 & -2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 3 \end{array}\right] \right\}$$

#### Example 25 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &-& 2 \, x_{2} &-& 8 \, x_{3} &+& 4 \, x_{4} &=& -5 \\ & & x_{2} &+& 2 \, x_{3} &-& 4 \, x_{4} &=& 4 \\ & & 2 \, x_{2} &+& 4 \, x_{3} &-& 7 \, x_{4} &=& 7 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & -2 & -8 & 4 & -5 \\ 0 & 1 & 2 & -4 & 4 \\ 0 & 2 & 4 & -7 & 7 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -4 & 0 & -1 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 4 \, a - 1 \\ -2 \, a \\ a \\ -1 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 26 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -1 \\ 0 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -1 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 1 \\ -2 \\ -4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} -1 & -1 & 1 & -2 & 1 \\ 0 & 1 & -1 & 1 & -2 \\ -2 & 0 & -1 & 0 & -4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & -2 & 2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -a + 1 \\ a \\ 2 \, a + 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 27 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -3 \\ 1 \\ 6 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -2 \\ -1 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} -2 & -4 & -4 & 0 \\ -2 & -3 & -2 & 2 \\ 0 & 1 & -1 & -4 \\ 3 & 6 & 4 & -4 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 2 \end{array}\right] \right\}$$

#### Example 28 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -3 \\ -2 \\ 2 \end{array}\right] = \left[\begin{array}{c} 7 \\ 7 \\ 5 \\ -5 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} -2 & -1 & -3 & 7 \\ 0 & 1 & -3 & 7 \\ 0 & 1 & -2 & 5 \\ 1 & 0 & 2 & -5 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ -2 \end{array}\right] \right\}$$

#### Example 29 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &+& x_{2} &-& 6 \, x_{3} &+& x_{4} &=& -3 \\ & & x_{2} &-& 4 \, x_{3} &-& 4 \, x_{4} &=& 3 \\-x_{1} &+& x_{2} &-& 2 \, x_{3} &-& 8 \, x_{4} &=& 8 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 1 & -6 & 1 & -3 \\ 0 & 1 & -4 & -4 & 3 \\ -1 & 1 & -2 & -8 & 8 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -2 & 0 & -1 \\ 0 & 1 & -4 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 1 \\ 4 \, a - 1 \\ a \\ -1 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 30 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 5 \\ -4 \\ 5 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 1 \\ -4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 1 \\ -4 \\ -4 \end{array}\right] = \left[\begin{array}{c} 6 \\ -7 \\ 6 \\ 4 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 5 & -4 & -4 & 6 \\ -4 & 1 & 1 & -7 \\ 5 & -4 & -4 & 6 \\ 4 & -4 & -4 & 4 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \\ -a + 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 31 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &-& 4 \, x_{2} &+& 6 \, x_{3} &+& 6 \, x_{4} &=& -2 \\ & & & & x_{3} &+& x_{4} &=& 0 \\ & & &-& 4 \, x_{3} &-& 4 \, x_{4} &=& 0 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & -4 & 6 & 6 & -2 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & -4 & -4 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & -4 & 0 & 0 & -2 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 4 \, a - 2 \\ a \\ -b \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 32 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} &-& x_{2} &-& x_{3} &=& -2 \\x_{1} &-& x_{2} & & &=& -5 \\-x_{1} &-& 2 \, x_{2} &-& 3 \, x_{3} &=& -1 \\ &-& 3 \, x_{2} &-& 3 \, x_{3} &=& -6 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 0 & -1 & -1 & -2 \\ 1 & -1 & 0 & -5 \\ -1 & -2 & -3 & -1 \\ 0 & -3 & -3 & -6 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 1 & -3 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -a - 3 \\ -a + 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 33 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} & & &-& x_{3} &=& 3 \\5 \, x_{1} &+& x_{2} & & &=& 8 \\-x_{1} &-& x_{2} &-& 3 \, x_{3} &=& 3 \\-2 \, x_{1} &-& x_{2} &+& x_{3} &=& -3 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 0 & -1 & 3 \\ 5 & 1 & 0 & 8 \\ -1 & -1 & -3 & 3 \\ -2 & -1 & 1 & -3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ -1 \end{array}\right] \right\}$$

#### Example 34 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &-& 5 \, x_{2} &-& 3 \, x_{3} &=& -6 \\-x_{1} &+& 6 \, x_{2} &+& 4 \, x_{3} &=& 7 \\x_{1} &-& 4 \, x_{2} &-& 2 \, x_{3} &=& -5 \\2 \, x_{1} &-& 5 \, x_{2} &-& x_{3} &=& -7 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -5 & -3 & -6 \\ -1 & 6 & 4 & 7 \\ 1 & -4 & -2 & -5 \\ 2 & -5 & -1 & -7 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 2 & -1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \, a - 1 \\ -a + 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 35 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} -x_{1} &-& x_{2} &+& 5 \, x_{3} &-& 3 \, x_{4} &=& -5 \\-x_{1} & & &+& 4 \, x_{3} &-& 2 \, x_{4} &=& -3 \\x_{1} &+& x_{2} &-& 6 \, x_{3} &+& 4 \, x_{4} &=& 6 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} -1 & -1 & 5 & -3 & -5 \\ -1 & 0 & 4 & -2 & -3 \\ 1 & 1 & -6 & 4 & 6 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & -1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & -1 & -1 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 1 \\ 1 \\ a - 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 36 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} & & &+& 3 \, x_{3} &+& 3 \, x_{4} &=& -5 \\2 \, x_{1} &+& x_{2} &+& 8 \, x_{3} &+& 4 \, x_{4} &=& -8 \\-x_{1} &-& x_{2} &-& 5 \, x_{3} & & &=& 1 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 0 & 3 & 3 & -5 \\ 2 & 1 & 8 & 4 & -8 \\ -1 & -1 & -5 & 0 & 1 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 3 & 0 & 1 \\ 0 & 1 & 2 & 0 & -2 \\ 0 & 0 & 0 & 1 & -2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \, a + 1 \\ -2 \, a - 2 \\ a \\ -2 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 37 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} 3 \, x_{1} &-& 3 \, x_{2} &+& 7 \, x_{3} &=& -5 \\3 \, x_{1} &-& 2 \, x_{2} &+& 4 \, x_{3} &=& -7 \\-x_{1} & & &+& 3 \, x_{3} &=& 6 \\-x_{1} &+& x_{2} &-& 2 \, x_{3} &=& 2 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 3 & -3 & 7 & -5 \\ 3 & -2 & 4 & -7 \\ -1 & 0 & 3 & 6 \\ -1 & 1 & -2 & 2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -3 \\ 1 \\ 1 \end{array}\right] \right\}$$

#### Example 38 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 0 \\ 0 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 0 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 1 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 2 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ -1 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 0 & 0 & 2 & 4 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ -1 & 4 & 0 & -3 & -1 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & -4 & 0 & 3 & 1 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 4 \, a - 3 \, b + 1 \\ a \\ -2 \, b \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 39 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} -3 \, x_{1} &-& 5 \, x_{2} &-& 2 \, x_{3} &=& 2 \\-3 \, x_{1} &-& 5 \, x_{2} &-& 4 \, x_{3} &=& 6 \\-x_{1} &-& x_{2} &+& x_{3} &=& -2 \\-4 \, x_{1} &-& 7 \, x_{2} &-& 4 \, x_{3} &=& 5 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} -3 & -5 & -2 & 2 \\ -3 & -5 & -4 & 6 \\ -1 & -1 & 1 & -2 \\ -4 & -7 & -4 & 5 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ -2 \end{array}\right] \right\}$$

#### Example 40 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &+& 2 \, x_{2} &-& 3 \, x_{3} &-& x_{4} &=& 7 \\ & & x_{2} &-& 2 \, x_{3} &+& x_{4} &=& 5 \\x_{1} &+& 2 \, x_{2} &-& 2 \, x_{3} &-& 2 \, x_{4} &=& 5 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 2 & -3 & -1 & 7 \\ 0 & 1 & -2 & 1 & 5 \\ 1 & 2 & -2 & -2 & 5 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & -1 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & -1 & -2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a - 1 \\ a + 1 \\ a - 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 41 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 4 \\ -3 \\ -1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -3 \\ -1 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -8 \\ -6 \\ 1 \end{array}\right] = \left[\begin{array}{c} -4 \\ -5 \\ -5 \\ 2 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 4 & 4 & 0 & -4 \\ -3 & -3 & -8 & -5 \\ -1 & -1 & -6 & -5 \\ -1 & -1 & 1 & 2 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -a - 1 \\ a \\ 1 \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 42 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &-& 2 \, x_{2} &+& 6 \, x_{3} &=& -4 \\ & & x_{2} &-& 2 \, x_{3} &=& 1 \\x_{1} &-& x_{2} &+& 4 \, x_{3} &=& -3 \\x_{1} &-& 3 \, x_{2} &+& 8 \, x_{3} &=& -5 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -2 & 6 & -4 \\ 0 & 1 & -2 & 1 \\ 1 & -1 & 4 & -3 \\ 1 & -3 & 8 & -5 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 2 & -2 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \, a - 2 \\ 2 \, a + 1 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 43 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 0 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ 2 \\ 2 \end{array}\right] = \left[\begin{array}{c} -1 \\ -2 \\ 1 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 1 & -2 & 5 & -1 \\ -1 & 0 & 0 & 2 & -2 \\ 1 & 1 & -1 & 2 & 1 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & 2 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & -3 & 2 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 2 \, a + 2 \\ -a + 1 \\ 3 \, a + 2 \\ a \end{array}\right] \middle|\,a\in\mathbb{R}\right\}$$

#### Example 44 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ -5 \\ 4 \\ -3 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -1 & 0 & 0 \\ -2 & -1 & 1 & -5 \\ 1 & 1 & -1 & 4 \\ 0 & -1 & 1 & -3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ -2 \end{array}\right] \right\}$$

#### Example 45 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &-& 5 \, x_{2} &-& 6 \, x_{3} &=& -7 \\ & & x_{2} &+& x_{3} &=& 0 \\x_{1} &-& 8 \, x_{2} &-& 8 \, x_{3} &=& -2 \\2 \, x_{1} &-& 4 \, x_{2} &-& 5 \, x_{3} &=& -8 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & -5 & -6 & -7 \\ 0 & 1 & 1 & 0 \\ 1 & -8 & -8 & -2 \\ 2 & -4 & -5 & -8 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

The solution set is $$\left\{\right\}$$

#### Example 46 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &+& 2 \, x_{2} &-& x_{3} &-& 7 \, x_{4} &=& 5 \\-x_{1} &-& x_{2} & & &+& 5 \, x_{4} &=& -3 \\ &-& 3 \, x_{2} &+& 3 \, x_{3} &+& 6 \, x_{4} &=& -6 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 2 & -1 & -7 & 5 \\ -1 & -1 & 0 & 5 & -3 \\ 0 & -3 & 3 & 6 & -6 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 1 & -3 & 1 \\ 0 & 1 & -1 & -2 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -a + 3 \, b + 1 \\ a + 2 \, b + 2 \\ a \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$

#### Example 47 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 4 \\ 0 \\ -1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 1 \\ -4 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 2 \\ -2 \\ 1 \end{array}\right] = \left[\begin{array}{c} -8 \\ -4 \\ 6 \\ 0 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 4 & 4 & 0 & -8 \\ 0 & 1 & 2 & -4 \\ -1 & -4 & -2 & 6 \\ -1 & 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} -2 \\ 0 \\ -2 \end{array}\right] \right\}$$

#### Example 48 🔗

Show how to find the solution set for the following vector equation

$x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ -1 \\ -1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ 2 \end{array}\right] = \left[\begin{array}{c} -6 \\ 6 \\ -4 \\ 6 \end{array}\right] .$

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 2 & -5 & -6 \\ -1 & -1 & 4 & 6 \\ 1 & -1 & -1 & -4 \\ -1 & 1 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

The solution set is $$\left\{ \left[\begin{array}{c} 0 \\ 2 \\ 2 \end{array}\right] \right\}$$

#### Example 49 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{4} x_{1} &+& 4 \, x_{2} &+& 7 \, x_{3} &=& -4 \\ & & x_{2} &+& 2 \, x_{3} &=& 0 \\-x_{1} &+& 2 \, x_{2} &+& 6 \, x_{3} &=& 6 \\ &-& 4 \, x_{2} &-& 8 \, x_{3} &=& 1 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{ccc|c} 1 & 4 & 7 & -4 \\ 0 & 1 & 2 & 0 \\ -1 & 2 & 6 & 6 \\ 0 & -4 & -8 & 1 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$

The solution set is $$\left\{\right\}$$

#### Example 50 🔗

Show how to find the solution set for the following system of linear equations. \begin{alignat*}{5} x_{1} &+& 4 \, x_{2} & & &+& 2 \, x_{4} &=& 3 \\ & & & & x_{3} &-& x_{4} &=& 2 \\-2 \, x_{1} &-& 8 \, x_{2} &-& x_{3} &-& 3 \, x_{4} &=& -8 \\ \end{alignat*}

$\mathrm{RREF} \left[\begin{array}{cccc|c} 1 & 4 & 0 & 2 & 3 \\ 0 & 0 & 1 & -1 & 2 \\ -2 & -8 & -1 & -3 & -8 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 4 & 0 & 2 & 3 \\ 0 & 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$
The solution set is $$\left\{ \left[\begin{array}{c} -4 \, a - 2 \, b + 3 \\ a \\ b + 2 \\ b \end{array}\right] \middle|\,a\text{\texttt{,}}b\in\mathbb{R}\right\}$$