## E1 - Linear systems, vector equations, and augmented matrices

#### Example 1 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} x & & &+& 2 \, z &+& 6 \, {w} &=& -6 \\ & & y & & &-& 2 \, {w} &=& -3 \\ & & 2 \, y &+& z &-& 2 \, {w} &=& -8 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 1 & 0 & 2 & 6 & -6 \\ 0 & 1 & 0 & -2 & -3 \\ 0 & 2 & 1 & -2 & -8 \end{array}\right]$

#### Example 2 π

Consider the system of equations \begin{alignat*}{4} x & & &-& 3 \, z &=& 4 \\ & & y &-& 3 \, z &=& -2 \\2 \, x &+& y &-& 8 \, z &=& 6 \\x &-& y &-& 3 \, z &=& 6 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 3 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

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

#### Example 4 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x &-& y &-& 5 \, z &=& 3 \\-x &-& 4 \, y &-& 5 \, z &=& 2 \\-x &-& y &+& z &=& -1 \\ &-& 2 \, y &-& 4 \, z &=& 2 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & -1 & -5 & 3 \\ -1 & -4 & -5 & 2 \\ -1 & -1 & 1 & -1 \\ 0 & -2 & -4 & 2 \end{array}\right]$

#### Example 5 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 6 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 7 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x_{1} &+& 3 \, x_{2} &-& 5 \, x_{3} &=& 3 \\3 \, x_{1} &+& x_{2} &-& 6 \, x_{3} &=& -6 \\ &-& x_{2} &+& x_{3} &=& -2 \\ &-& 2 \, x_{2} & & &=& -6 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 3 & -5 & 3 \\ 3 & 1 & -6 & -6 \\ 0 & -1 & 1 & -2 \\ 0 & -2 & 0 & -6 \end{array}\right]$

#### Example 8 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} 4 \, x &-& 3 \, y &-& 2 \, z &-& 8 \, {w} &=& 1 \\3 \, x &-& 2 \, y &-& z &-& 7 \, {w} &=& -1 \\2 \, x &-& 3 \, y &-& 3 \, z & & &=& 8 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 4 & -3 & -2 & -8 & 1 \\ 3 & -2 & -1 & -7 & -1 \\ 2 & -3 & -3 & 0 & 8 \end{array}\right]$

#### Example 9 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{5} x &-& 2 \, y & & &+& 3 \, {w} &=& 3 \\ & & y &+& 2 \, z &+& 2 \, {w} &=& -5 \\x &-& 2 \, y &+& z &+& 5 \, {w} &=& 1 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\right] = \left[\begin{array}{c} 3 \\ -5 \\ 1 \end{array}\right]$

#### Example 10 π

Consider the system of equations \begin{alignat*}{5} x &-& y &+& 3 \, z &-& 4 \, {w} &=& 0 \\ & & y &-& 4 \, z &+& 2 \, {w} &=& -1 \\ &-& y &+& 4 \, z &-& 2 \, {w} &=& 1 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 11 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} 2 \, x &+& 3 \, y &-& 3 \, z &+& 7 \, {w} &=& 4 \\x &+& 2 \, y &-& z &+& 5 \, {w} &=& 3 \\4 \, x &+& 4 \, y &-& 8 \, z &+& 8 \, {w} &=& 4 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 2 & 3 & -3 & 7 & 4 \\ 1 & 2 & -1 & 5 & 3 \\ 4 & 4 & -8 & 8 & 4 \end{array}\right]$

#### Example 12 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 13 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 14 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x_{1} &+& x_{2} &+& 4 \, x_{3} &=& 1 \\ & & x_{2} &-& x_{3} &=& 2 \\ &-& 4 \, x_{2} &+& 4 \, x_{3} &=& -7 \\ & & 4 \, x_{2} &-& 4 \, x_{3} &=& 7 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 1 & 4 & 1 \\ 0 & 1 & -1 & 2 \\ 0 & -4 & 4 & -7 \\ 0 & 4 & -4 & 7 \end{array}\right]$

#### Example 15 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 16 π

Consider the system of equations \begin{alignat*}{4} 6 \, x &+& 5 \, y &-& 7 \, z &=& -4 \\-5 \, x &-& 4 \, y &+& 6 \, z &=& 4 \\5 \, x &+& 3 \, y &-& 6 \, z &=& -7 \\-5 \, x &-& 3 \, y &+& 7 \, z &=& 8 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

1. $\left[\begin{array}{ccc|c} 6 & 5 & -7 & -4 \\ -5 & -4 & 6 & 4 \\ 5 & 3 & -6 & -7 \\ -5 & -3 & 7 & 8 \end{array}\right]$

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

#### Example 17 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x &+& 2 \, y &+& 5 \, z &=& 8 \\ & & & & z &=& 2 \\ & & & & z &=& 2 \\-x &-& 2 \, y &-& z &=& 0 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 2 & 5 & 8 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ -1 & -2 & -1 & 0 \end{array}\right]$

#### Example 18 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{4} -x_{1} &-& x_{2} &+& x_{3} &=& -3 \\3 \, x_{1} &-& 2 \, x_{2} &-& 4 \, x_{3} &=& 7 \\3 \, x_{1} &-& 4 \, x_{2} &-& x_{3} &=& -4 \\-2 \, x_{1} & & &+& 2 \, x_{3} &=& -4 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} -1 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -4 \\ -1 \\ 2 \end{array}\right] = \left[\begin{array}{c} -3 \\ 7 \\ -4 \\ -4 \end{array}\right]$

#### Example 19 π

Consider the system of equations \begin{alignat*}{4} x &+& 5 \, y & & &=& 0 \\-x &-& 4 \, y & & &=& 0 \\ &-& y &+& z &=& 3 \\-2 \, x &-& 5 \, y &-& 2 \, z &=& -6 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 20 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{4} -7 \, x &-& 5 \, y &+& z &=& -5 \\-4 \, x &-& 3 \, y & & &=& -2 \\-2 \, x &-& y &+& 3 \, z &=& -6 \\5 \, x &+& 3 \, y & & &=& 1 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} -7 \\ -4 \\ -2 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 0 \\ 3 \\ 0 \end{array}\right] = \left[\begin{array}{c} -5 \\ -2 \\ -6 \\ 1 \end{array}\right]$

#### Example 21 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} x &+& 2 \, y &+& 3 \, z &+& 4 \, {w} &=& -3 \\ & & y &+& z &+& {w} &=& -1 \\ & & 2 \, y &+& 3 \, z &+& 4 \, {w} &=& -4 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 1 & 2 & 3 & 4 & -3 \\ 0 & 1 & 1 & 1 & -1 \\ 0 & 2 & 3 & 4 & -4 \end{array}\right]$

#### Example 22 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{4} 4 \, x &-& 6 \, y &-& 7 \, z &=& 0 \\x &-& 2 \, y &-& z &=& -3 \\-x &+& y &+& 2 \, z &=& -2 \\-2 \, x &+& 5 \, y &+& 3 \, z &=& 7 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ -2 \\ 1 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} -7 \\ -1 \\ 2 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ -3 \\ -2 \\ 7 \end{array}\right]$

#### Example 23 π

Consider the system of equations \begin{alignat*}{5} -x &-& 2 \, y &+& 5 \, z &-& 3 \, {w} &=& -1 \\2 \, x &+& 3 \, y &-& 8 \, z &+& 4 \, {w} &=& 2 \\4 \, x &+& y &-& 6 \, z &-& 2 \, {w} &=& 4 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

1. $\left[\begin{array}{cccc|c} -1 & -2 & 5 & -3 & -1 \\ 2 & 3 & -8 & 4 & 2 \\ 4 & 1 & -6 & -2 & 4 \end{array}\right]$

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

#### Example 24 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{5} x &-& 2 \, y &+& z &+& {w} &=& 1 \\ & & & & z &+& 2 \, {w} &=& 1 \\x &-& 2 \, y &+& 4 \, z &+& 7 \, {w} &=& 4 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 2 \\ 7 \end{array}\right] = \left[\begin{array}{c} 1 \\ 1 \\ 4 \end{array}\right]$

#### Example 25 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x_{1} & & &-& 2 \, x_{3} &=& 2 \\ & & x_{2} & & &=& 2 \\-x_{1} & & &+& 2 \, x_{3} &=& -2 \\x_{1} &+& x_{2} &-& 2 \, x_{3} &=& 4 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 0 & -2 & 2 \\ 0 & 1 & 0 & 2 \\ -1 & 0 & 2 & -2 \\ 1 & 1 & -2 & 4 \end{array}\right]$

#### Example 26 π

Consider the system of equations \begin{alignat*}{5} 2 \, x_{1} &-& 4 \, x_{2} &-& 5 \, x_{3} &+& 4 \, x_{4} &=& -5 \\-x_{1} &+& 2 \, x_{2} &+& 3 \, x_{3} &-& 2 \, x_{4} &=& 3 \\x_{1} &-& 3 \, x_{2} &-& 6 \, x_{3} &+& 2 \, x_{4} &=& -6 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 27 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 28 π

Consider the system of equations \begin{alignat*}{5} &-& x_{2} &+& 3 \, x_{3} &+& 6 \, x_{4} &=& -3 \\x_{1} & & &+& 4 \, x_{3} &+& 3 \, x_{4} &=& 1 \\-x_{1} & & &-& 4 \, x_{3} &-& 2 \, x_{4} &=& -1 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 29 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{4} -2 \, x &+& y &+& 6 \, z &=& 0 \\x &-& y &-& 4 \, z &=& 0 \\x &-& y &-& 4 \, z &=& 0 \\ &-& 3 \, y &-& 6 \, z &=& 0 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -1 \\ -1 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 6 \\ -4 \\ -4 \\ -6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$

#### Example 30 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} x_{1} &-& 4 \, x_{2} &+& 3 \, x_{3} & & &=& -2 \\ & & & & & & x_{4} &=& 3 \\-x_{1} &+& 4 \, x_{2} &-& 3 \, x_{3} &-& 2 \, x_{4} &=& -4 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 1 & -4 & 3 & 0 & -2 \\ 0 & 0 & 0 & 1 & 3 \\ -1 & 4 & -3 & -2 & -4 \end{array}\right]$

#### Example 31 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x &+& 2 \, y & & &=& 7 \\-x &-& 3 \, y &+& 4 \, z &=& 2 \\ & & 3 \, y &-& 5 \, z &=& -6 \\ & & y &-& 3 \, z &=& -6 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 2 & 0 & 7 \\ -1 & -3 & 4 & 2 \\ 0 & 3 & -5 & -6 \\ 0 & 1 & -3 & -6 \end{array}\right]$

#### Example 32 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 33 π

Consider the system of equations \begin{alignat*}{4} &-& x_{2} & & &=& 0 \\ & & x_{2} &+& 2 \, x_{3} &=& -6 \\x_{1} &+& x_{2} &+& x_{3} &=& -4 \\-x_{1} &-& 3 \, x_{2} &-& 2 \, x_{3} &=& 7 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 34 π

Consider the system of equations \begin{alignat*}{5} x_{1} &-& x_{2} &-& 6 \, x_{3} &-& 8 \, x_{4} &=& -3 \\ & & x_{2} &+& 3 \, x_{3} &+& 4 \, x_{4} &=& 3 \\ & & & & x_{3} &+& x_{4} &=& 1 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 35 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 36 π

Consider the augmented matrix

$\left[\begin{array}{ccc|c} -1 & 0 & 5 & -6 \\ -2 & -7 & -8 & 6 \\ -1 & -2 & 0 & -1 \\ 1 & 3 & 8 & -7 \end{array}\right]$

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 37 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 38 π

Consider the system of equations \begin{alignat*}{5} -x &-& y &-& 2 \, z &+& 3 \, {w} &=& 3 \\-x &+& 2 \, y &-& z &-& 5 \, {w} &=& 6 \\2 \, x & & &+& 3 \, z &-& {w} &=& -7 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 39 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} 2 \, x_{1} &-& 6 \, x_{2} &+& x_{3} & & &=& -6 \\x_{1} & & &+& 3 \, x_{3} &+& 5 \, x_{4} &=& -5 \\x_{1} &-& 5 \, x_{2} &-& x_{3} &-& 3 \, x_{4} &=& -2 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 2 & -6 & 1 & 0 & -6 \\ 1 & 0 & 3 & 5 & -5 \\ 1 & -5 & -1 & -3 & -2 \end{array}\right]$

#### Example 40 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{5} x_{1} &+& 3 \, x_{2} &+& 3 \, x_{3} & & &=& -3 \\-3 \, x_{1} &-& 5 \, x_{2} &-& 6 \, x_{3} &+& x_{4} &=& 3 \\ & & x_{2} &+& x_{3} & & &=& -1 \\ \end{alignat*}
2. $\left[\begin{array}{cccc|c} 1 & 3 & 3 & 0 & -3 \\ -3 & -5 & -6 & 1 & 3 \\ 0 & 1 & 1 & 0 & -1 \end{array}\right]$

#### Example 41 π

Consider the system of equations \begin{alignat*}{5} x_{1} &+& 5 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 0 \\x_{1} &+& 5 \, x_{2} & & &+& x_{4} &=& 2 \\x_{1} &+& 5 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 0 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 42 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

1. \begin{alignat*}{5} x &-& y & & &+& 2 \, {w} &=& 6 \\ & & y &-& 2 \, z &-& {w} &=& -4 \\ & & y &-& 2 \, z & & &=& -3 \\ \end{alignat*}
2. $x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -2 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 2 \\ -1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 6 \\ -4 \\ -3 \end{array}\right]$

#### Example 43 π

Consider the system of equations \begin{alignat*}{4} -x &+& y &-& z &=& -4 \\-x & & &-& 4 \, z &=& -5 \\x &-& y &+& 2 \, z &=& 5 \\-x & & &-& 7 \, z &=& -8 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 44 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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

#### Example 45 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} x_{1} &+& 5 \, x_{2} &-& 3 \, x_{3} &=& 3 \\ & & & & x_{3} &=& -1 \\-x_{1} &-& 5 \, x_{2} &+& 4 \, x_{3} &=& -4 \\x_{1} &+& 5 \, x_{2} & & &=& 0 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 1 & 5 & -3 & 3 \\ 0 & 0 & 1 & -1 \\ -1 & -5 & 4 & -4 \\ 1 & 5 & 0 & 0 \end{array}\right]$

#### Example 46 π

Consider the system of equations \begin{alignat*}{5} x & & &-& 2 \, z &+& {w} &=& -8 \\x &+& y &-& 2 \, z &+& 3 \, {w} &=& -6 \\ &-& 4 \, y &+& z &-& 8 \, {w} &=& -4 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 47 π

Consider the system of equations \begin{alignat*}{5} -x &+& y &+& 5 \, z &+& 3 \, {w} &=& 8 \\-x & & &+& z & & &=& 2 \\-3 \, x &-& 2 \, y &-& 4 \, z &-& 5 \, {w} &=& -5 \\ \end{alignat*}

1. Write an augmented matrix corresponding to this system.
2. Write a vector equation corresponding to this system.

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

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

#### Example 48 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

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

#### Example 49 π

Consider the vector equation.

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

1. Write a system of scalar equations corresponding to this vector equation.
2. Write an augmented matrix corresponding to this vector equation.

1. \begin{alignat*}{4} & & 2 \, x_{2} &+& 2 \, x_{3} &=& 4 \\-x_{1} &-& 3 \, x_{2} &-& 4 \, x_{3} &=& -6 \\ &-& x_{2} & & &=& -1 \\-x_{1} &-& 2 \, x_{2} &-& 3 \, x_{3} &=& -4 \\ \end{alignat*}
2. $\left[\begin{array}{ccc|c} 0 & 2 & 2 & 4 \\ -1 & -3 & -4 & -6 \\ 0 & -1 & 0 & -1 \\ -1 & -2 & -3 & -4 \end{array}\right]$

#### Example 50 π

Consider the augmented matrix

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

1. Write a system of scalar equations corresponding to this augmented matrix.
2. Write a vector equation corresponding to this augmented matrix.

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