E3 - Solving systems


Example 1

E3 - Solving systems (ver. 1)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} - 6 \, x_{3} &= 8 \\ x_{2} - 5 \, x_{3} &= 6 \\ -2 \, x_{1} + x_{2} + 4 \, x_{3} &= -5 \\ -x_{2} + 2 \, x_{3} &= -3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 1 & -6 & 8 \\ 0 & 1 & -5 & 6 \\ -2 & 1 & 4 & -5 \\ 0 & -1 & 2 & -3 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right] \right\}\]


Example 2

E3 - Solving systems (ver. 2)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} - 3 \, x_{3} &= 3 \\ 2 \, x_{1} + x_{2} + 6 \, x_{3} &= -4 \\ -2 \, x_{1} - 7 \, x_{3} &= 6 \\ 2 \, x_{1} + 4 \, x_{2} + 3 \, x_{3} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -1 & 0 & -3 & 3 \\ 2 & 1 & 6 & -4 \\ -2 & 0 & -7 & 6 \\ 2 & 4 & 3 & 2 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -3 \\ 2 \\ 0 \end{array}\right] \right\}\]


Example 3

E3 - Solving systems (ver. 3)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 4 \, x_{2} + x_{4} &= 5 \\ x_{3} - 3 \, x_{4} &= -6 \\ x_{3} - 2 \, x_{4} &= -4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 4 & 0 & 1 & 5 \\ 0 & 0 & 1 & -3 & -6 \\ 0 & 0 & 1 & -2 & -4 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 4 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -4 \, a + 3 \\ a \\ 0 \\ 2 \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 4

E3 - Solving systems (ver. 4)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{2} &= 2 \\ x_{1} + x_{3} &= -2 \\ 2 \, x_{1} + 2 \, x_{3} &= -4 \\ -3 \, x_{1} + 7 \, x_{2} - 3 \, x_{3} &= -8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 0 & -1 & 0 & 2 \\ 1 & 0 & 1 & -2 \\ 2 & 0 & 2 & -4 \\ -3 & 7 & -3 & -8 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 1 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a - 2 \\ -2 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 5

E3 - Solving systems (ver. 5)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} - 3 \, x_{3} - 3 \, x_{4} &= -7 \\ x_{2} - 3 \, x_{3} - 4 \, x_{4} &= -5 \\ x_{1} + x_{2} - 2 \, x_{3} - 3 \, x_{4} &= -5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 1 & -3 & -3 & -7 \\ 0 & 1 & -3 & -4 & -5 \\ 1 & 1 & -2 & -3 & -5 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 1 & -2 \\ 0 & 1 & 0 & -4 & 1 \\ 0 & 0 & 1 & 0 & 2 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a - 2 \\ 4 \, a + 1 \\ 2 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 6

E3 - Solving systems (ver. 6)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 2 \, x_{2} + x_{3} &= 6 \\ x_{1} - x_{2} - 2 \, x_{3} &= -2 \\ x_{2} + 5 \, x_{3} &= -3 \\ -2 \, x_{1} + 4 \, x_{2} + 7 \, x_{3} &= 6 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -1 & 2 & 1 & 6 \\ 1 & -1 & -2 & -2 \\ 0 & 1 & 5 & -3 \\ -2 & 4 & 7 & 6 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 7

E3 - Solving systems (ver. 7)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 4 \, x_{1} - 6 \, x_{2} + 3 \, x_{3} &= 5 \\ -x_{1} + 4 \, x_{2} + x_{3} &= 8 \\ -x_{1} + x_{2} - x_{3} &= -3 \\ -3 \, x_{1} + 6 \, x_{2} + x_{3} &= 5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 4 & -6 & 3 & 5 \\ -1 & 4 & 1 & 8 \\ -1 & 1 & -1 & -3 \\ -3 & 6 & 1 & 5 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 8

E3 - Solving systems (ver. 8)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -2 \, x_{1} - x_{2} &= 3 \\ 3 \, x_{1} + x_{2} + x_{3} &= -2 \\ x_{1} + 2 \, x_{2} - 2 \, x_{3} &= -4 \\ x_{3} &= 6 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -2 & -1 & 0 & 3 \\ 3 & 1 & 1 & -2 \\ 1 & 2 & -2 & -4 \\ 0 & 0 & 1 & 6 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 9

E3 - Solving systems (ver. 9)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - 5 \, x_{2} &= 1 \\ x_{3} &= 1 \\ x_{1} - 5 \, x_{2} + 5 \, x_{3} &= 6 \\ x_{1} - 5 \, x_{2} + 3 \, x_{3} &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -5 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & -5 & 5 & 6 \\ 1 & -5 & 3 & 4 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & -5 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 5 \, a + 1 \\ a \\ 1 \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 10

E3 - Solving systems (ver. 10)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} + 4 \, x_{3} &= -6 \\ x_{1} &= 1 \\ x_{2} + 3 \, x_{3} &= -6 \\ -x_{2} - 2 \, x_{3} &= 5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 1 & 4 & -6 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 3 & -6 \\ 0 & -1 & -2 & 5 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 1 \\ -3 \\ -1 \end{array}\right] \right\}\]


Example 11

E3 - Solving systems (ver. 11)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 5 \, x_{2} + 5 \, x_{3} + 8 \, x_{4} &= -6 \\ x_{4} &= -1 \\ x_{1} - 5 \, x_{2} - 5 \, x_{3} - 3 \, x_{4} &= 1 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 5 & 5 & 8 & -6 \\ 0 & 0 & 0 & 1 & -1 \\ 1 & -5 & -5 & -3 & 1 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & -5 & -5 & 0 & -2 \\ 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 5 \, a + 5 \, b - 2 \\ a \\ b \\ -1 \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 12

E3 - Solving systems (ver. 12)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 3 \, x_{1} - 8 \, x_{2} - 7 \, x_{3} - 5 \, x_{4} &= -8 \\ 2 \, x_{1} - 5 \, x_{2} - 4 \, x_{3} - 4 \, x_{4} &= -5 \\ 3 \, x_{2} + 7 \, x_{3} - 8 \, x_{4} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 3 & -8 & -7 & -5 & -8 \\ 2 & -5 & -4 & -4 & -5 \\ 0 & 3 & 7 & -8 & 3 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 2 & 1 \\ 0 & 0 & 1 & -2 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a \\ -2 \, a + 1 \\ 2 \, a \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 13

E3 - Solving systems (ver. 13)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{3} - x_{4} &= -4 \\ x_{2} + 2 \, x_{3} - 2 \, x_{4} &= -7 \\ -x_{1} - x_{3} + 3 \, x_{4} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 0 & 0 & 1 & -1 & -4 \\ 0 & 1 & 2 & -2 & -7 \\ -1 & 0 & -1 & 3 & 3 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -2 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & -1 & -4 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 2 \, a + 1 \\ 1 \\ a - 4 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 14

E3 - Solving systems (ver. 14)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 4 \, x_{2} + 7 \, x_{3} + 7 \, x_{4} &= 4 \\ -x_{1} + 3 \, x_{2} + 6 \, x_{3} + 5 \, x_{4} &= 3 \\ -2 \, x_{1} + 2 \, x_{2} + 8 \, x_{3} + 2 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 4 & 7 & 7 & 4 \\ -1 & 3 & 6 & 5 & 3 \\ -2 & 2 & 8 & 2 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & -3 & 1 & 0 \\ 0 & 1 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 3 \, a - b \\ -a - 2 \, b + 1 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 15

E3 - Solving systems (ver. 15)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -5 \, x_{1} + 4 \, x_{2} + x_{3} &= -5 \\ -x_{1} + 3 \, x_{2} + 4 \, x_{3} &= -8 \\ -2 \, x_{1} + 2 \, x_{2} + x_{3} &= -3 \\ -2 \, x_{1} + 5 \, x_{2} + 4 \, x_{3} &= -5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -5 & 4 & 1 & -5 \\ -1 & 3 & 4 & -8 \\ -2 & 2 & 1 & -3 \\ -2 & 5 & 4 & -5 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 16

E3 - Solving systems (ver. 16)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + x_{2} + 3 \, x_{3} - 3 \, x_{4} &= -7 \\ -x_{1} + 2 \, x_{4} &= -2 \\ -x_{2} - 2 \, x_{3} + 3 \, x_{4} &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 1 & 3 & -3 & -7 \\ -1 & 0 & 0 & 2 & -2 \\ 0 & -1 & -2 & 3 & 4 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -2 & 2 \\ 0 & 1 & 0 & 1 & -2 \\ 0 & 0 & 1 & -2 & -1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 2 \, a + 2 \\ -a - 2 \\ 2 \, a - 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 17

E3 - Solving systems (ver. 17)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 2 \, x_{3} - x_{4} &= -1 \\ x_{1} - x_{2} - 3 \, x_{3} + 4 \, x_{4} &= -2 \\ -2 \, x_{1} + 2 \, x_{2} + 6 \, x_{3} - 8 \, x_{4} &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 0 & 2 & -1 & -1 \\ 1 & -1 & -3 & 4 & -2 \\ -2 & 2 & 6 & -8 & 4 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & -2 & 1 & 1 \\ 0 & 1 & 1 & -3 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 2 \, a - b + 1 \\ -a + 3 \, b + 3 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 18

E3 - Solving systems (ver. 18)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -5 \, x_{1} + 5 \, x_{2} + 8 \, x_{3} &= -2 \\ 5 \, x_{1} - 4 \, x_{2} - 4 \, x_{3} &= 5 \\ 2 \, x_{1} - x_{2} - 3 \, x_{3} &= 0 \\ -3 \, x_{1} + 3 \, x_{2} + 4 \, x_{3} &= -2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -5 & 5 & 8 & -2 \\ 5 & -4 & -4 & 5 \\ 2 & -1 & -3 & 0 \\ -3 & 3 & 4 & -2 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right] \right\}\]


Example 19

E3 - Solving systems (ver. 19)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 2 \, x_{2} + x_{3} + 2 \, x_{4} &= -3 \\ 2 \, x_{1} - 3 \, x_{2} - 3 \, x_{3} - 8 \, x_{4} &= 4 \\ x_{1} - 2 \, x_{2} - 2 \, x_{3} - 5 \, x_{4} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 2 & 1 & 2 & -3 \\ 2 & -3 & -3 & -8 & 4 \\ 1 & -2 & -2 & -5 & 3 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1 & -1 \\ 0 & 1 & 0 & -1 & -2 \\ 0 & 0 & 1 & 3 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a - 1 \\ a - 2 \\ -3 \, a \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 20

E3 - Solving systems (ver. 20)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - 2 \, x_{3} + 6 \, x_{4} &= -5 \\ x_{1} + x_{2} - x_{3} + 3 \, x_{4} &= 0 \\ -x_{1} + 3 \, x_{3} - 8 \, x_{4} &= 8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -2 & 6 & -5 \\ 1 & 1 & -1 & 3 & 0 \\ -1 & 0 & 3 & -8 & 8 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 2 & 1 \\ 0 & 1 & 0 & -1 & 2 \\ 0 & 0 & 1 & -2 & 3 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \, a + 1 \\ a + 2 \\ 2 \, a + 3 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 21

E3 - Solving systems (ver. 21)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 3 \, x_{2} - x_{3} - 6 \, x_{4} &= -2 \\ x_{2} - x_{3} - x_{4} &= 0 \\ -x_{1} - x_{2} - x_{3} + 4 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 3 & -1 & -6 & -2 \\ 0 & 1 & -1 & -1 & 0 \\ -1 & -1 & -1 & 4 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 2 & -3 & -2 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \, a + 3 \, b - 2 \\ a + b \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 22

E3 - Solving systems (ver. 22)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} - x_{2} - 2 \, x_{3} - 3 \, x_{4} &= 2 \\ -x_{1} - 2 \, x_{2} - 3 \, x_{3} - 5 \, x_{4} &= 3 \\ 2 \, x_{1} + 3 \, x_{2} + 5 \, x_{3} + 8 \, x_{4} &= -5 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & -1 & -2 & -3 & 2 \\ -1 & -2 & -3 & -5 & 3 \\ 2 & 3 & 5 & 8 & -5 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 1 & 1 & -1 \\ 0 & 1 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a - b - 1 \\ -a - 2 \, b - 1 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 23

E3 - Solving systems (ver. 23)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 2 \, x_{2} + x_{3} - 2 \, x_{4} &= -3 \\ -x_{1} - 2 \, x_{2} + 2 \, x_{3} - 7 \, x_{4} &= -6 \\ 2 \, x_{3} - 6 \, x_{4} &= -6 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 2 & 1 & -2 & -3 \\ -1 & -2 & 2 & -7 & -6 \\ 0 & 0 & 2 & -6 & -6 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 2 & 0 & 1 & 0 \\ 0 & 0 & 1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \, a - b \\ a \\ 3 \, b - 3 \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 24

E3 - Solving systems (ver. 24)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 2 \, x_{1} + x_{2} + 3 \, x_{3} &= -1 \\ x_{1} + x_{2} + 3 \, x_{3} - x_{4} &= 2 \\ 3 \, x_{1} + x_{2} + 4 \, x_{3} + x_{4} &= -3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 2 & 1 & 3 & 0 & -1 \\ 1 & 1 & 3 & -1 & 2 \\ 3 & 1 & 4 & 1 & -3 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 1 & -3 \\ 0 & 1 & 0 & -2 & 2 \\ 0 & 0 & 1 & 0 & 1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a - 3 \\ 2 \, a + 2 \\ 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 25

E3 - Solving systems (ver. 25)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{3} - 3 \, x_{4} &= 2 \\ 2 \, x_{1} + x_{2} - 5 \, x_{4} &= -3 \\ -x_{1} - x_{2} + 2 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -1 & -3 & 2 \\ 2 & 1 & 0 & -5 & -3 \\ -1 & -1 & 0 & 2 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -3 & -1 \\ 0 & 1 & 0 & 1 & -1 \\ 0 & 0 & 1 & 0 & -3 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 3 \, a - 1 \\ -a - 1 \\ -3 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 26

E3 - Solving systems (ver. 26)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 5 \, x_{1} - 6 \, x_{2} - 3 \, x_{3} &= -6 \\ -x_{1} + x_{2} + x_{3} &= 3 \\ 2 \, x_{1} - 3 \, x_{2} - x_{3} &= -2 \\ 2 \, x_{1} + x_{2} - x_{3} &= 3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 5 & -6 & -3 & -6 \\ -1 & 1 & 1 & 3 \\ 2 & -3 & -1 & -2 \\ 2 & 1 & -1 & 3 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 27

E3 - Solving systems (ver. 27)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{2} - 4 \, x_{3} + 6 \, x_{4} &= 2 \\ x_{2} + 3 \, x_{3} - 5 \, x_{4} &= -1 \\ x_{1} - x_{2} - 3 \, x_{3} + 4 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -4 & 6 & 2 \\ 0 & 1 & 3 & -5 & -1 \\ 1 & -1 & -3 & 4 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1 & 1 \\ 0 & 1 & 0 & 1 & -1 \\ 0 & 0 & 1 & -2 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a + 1 \\ -a - 1 \\ 2 \, a \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 28

E3 - Solving systems (ver. 28)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{2} - 2 \, x_{3} - 4 \, x_{4} &= -3 \\ x_{2} - x_{3} &= -4 \\ -x_{1} + 4 \, x_{2} + 5 \, x_{4} &= -7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -2 & -4 & -3 \\ 0 & 1 & -1 & 0 & -4 \\ -1 & 4 & 0 & 5 & -7 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1 & -1 \\ 0 & 1 & 0 & 1 & -2 \\ 0 & 0 & 1 & 1 & 2 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a - 1 \\ -a - 2 \\ -a + 2 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 29

E3 - Solving systems (ver. 29)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - 5 \, x_{2} - 2 \, x_{3} &= 7 \\ x_{2} + x_{3} &= -3 \\ x_{2} + 2 \, x_{3} &= -6 \\ -2 \, x_{2} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -5 & -2 & 7 \\ 0 & 1 & 1 & -3 \\ 0 & 1 & 2 & -6 \\ 0 & -2 & 0 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 1 \\ 0 \\ -3 \end{array}\right] \right\}\]


Example 30

E3 - Solving systems (ver. 30)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{2} - 4 \, x_{3} - 2 \, x_{4} &= 7 \\ x_{2} + 5 \, x_{3} + 4 \, x_{4} &= -6 \\ x_{3} + x_{4} &= -1 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & -4 & -2 & 7 \\ 0 & 1 & 5 & 4 & -6 \\ 0 & 0 & 1 & 1 & -1 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & -1 & -1 \\ 0 & 0 & 1 & 1 & -1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a + 2 \\ a - 1 \\ -a - 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 31

E3 - Solving systems (ver. 31)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 4 \, x_{2} + 8 \, x_{3} - 3 \, x_{4} &= 6 \\ -x_{1} - 5 \, x_{3} - x_{4} &= -8 \\ -3 \, x_{2} - 2 \, x_{3} + 3 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 4 & 8 & -3 & 6 \\ -1 & 0 & -5 & -1 & -8 \\ 0 & -3 & -2 & 3 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 1 & -2 \\ 0 & 1 & 0 & -1 & -2 \\ 0 & 0 & 1 & 0 & 2 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a - 2 \\ a - 2 \\ 2 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 32

E3 - Solving systems (ver. 32)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 4 \, x_{1} + x_{2} + 7 \, x_{3} &= -5 \\ -x_{1} - 3 \, x_{3} &= 0 \\ x_{1} + 4 \, x_{3} &= 1 \\ x_{1} + 3 \, x_{3} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 4 & 1 & 7 & -5 \\ -1 & 0 & -3 & 0 \\ 1 & 0 & 4 & 1 \\ 1 & 0 & 3 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -3 \\ 0 \\ 1 \end{array}\right] \right\}\]


Example 33

E3 - Solving systems (ver. 33)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} + x_{3} + 2 \, x_{4} &= -4 \\ x_{2} + 2 \, x_{3} + x_{4} &= -2 \\ -x_{1} + 2 \, x_{2} + 5 \, x_{3} + x_{4} &= -2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 1 & 1 & 2 & -4 \\ 0 & 1 & 2 & 1 & -2 \\ -1 & 2 & 5 & 1 & -2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & -1 & 1 & -2 \\ 0 & 1 & 2 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a - b - 2 \\ -2 \, a - b - 2 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 34

E3 - Solving systems (ver. 34)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 2 \, x_{2} - x_{3} &= 3 \\ x_{1} + x_{2} &= 3 \\ -x_{2} &= -4 \\ -2 \, x_{1} + x_{3} &= 7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 2 & -1 & 3 \\ 1 & 1 & 0 & 3 \\ 0 & -1 & 0 & -4 \\ -2 & 0 & 1 & 7 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 35

E3 - Solving systems (ver. 35)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - 4 \, x_{2} + 4 \, x_{3} + x_{4} &= -8 \\ x_{1} - 3 \, x_{2} + 4 \, x_{3} + 3 \, x_{4} &= -6 \\ x_{3} + 2 \, x_{4} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -4 & 4 & 1 & -8 \\ 1 & -3 & 4 & 3 & -6 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 & 2 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -a \\ -2 \, a + 2 \\ -2 \, a \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 36

E3 - Solving systems (ver. 36)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{2} &= 1 \\ 3 \, x_{1} - 2 \, x_{2} + 2 \, x_{3} &= -5 \\ 3 \, x_{1} - 3 \, x_{2} + x_{3} &= 0 \\ -2 \, x_{1} + x_{2} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & -1 & 0 & 1 \\ 3 & -2 & 2 & -5 \\ 3 & -3 & 1 & 0 \\ -2 & 1 & 0 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -1 \\ -2 \\ -3 \end{array}\right] \right\}\]


Example 37

E3 - Solving systems (ver. 37)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} + 2 \, x_{2} + 7 \, x_{3} &= 0 \\ -x_{1} + x_{2} + 6 \, x_{3} &= 1 \\ 2 \, x_{1} + x_{2} - 8 \, x_{3} &= -5 \\ -2 \, x_{1} - x_{2} + 6 \, x_{3} &= 6 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} -1 & 2 & 7 & 0 \\ -1 & 1 & 6 & 1 \\ 2 & 1 & -8 & -5 \\ -2 & -1 & 6 & 6 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

\[\emptyset\]


Example 38

E3 - Solving systems (ver. 38)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} + x_{3} - 2 \, x_{4} &= -1 \\ x_{2} + 2 \, x_{3} - x_{4} &= 1 \\ 0 &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 1 & 1 & -2 & -1 \\ 0 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & -1 & -1 & -2 \\ 0 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a + b - 2 \\ -2 \, a + b + 1 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 39

E3 - Solving systems (ver. 39)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 3 \, x_{1} - 8 \, x_{2} - 6 \, x_{3} &= -4 \\ -2 \, x_{1} + x_{2} + 4 \, x_{3} &= 7 \\ -2 \, x_{1} + 3 \, x_{2} + 4 \, x_{3} &= 5 \\ -x_{1} + 4 \, x_{2} + 2 \, x_{3} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 3 & -8 & -6 & -4 \\ -2 & 1 & 4 & 7 \\ -2 & 3 & 4 & 5 \\ -1 & 4 & 2 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & -2 & -4 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 2 \, a - 4 \\ -1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 40

E3 - Solving systems (ver. 40)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 2 \, x_{1} + 8 \, x_{2} - 2 \, x_{4} &= 4 \\ x_{1} + 4 \, x_{2} + 3 \, x_{3} - 4 \, x_{4} &= 8 \\ x_{1} + 4 \, x_{2} - 2 \, x_{3} + x_{4} &= -2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 2 & 8 & 0 & -2 & 4 \\ 1 & 4 & 3 & -4 & 8 \\ 1 & 4 & -2 & 1 & -2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 4 & 0 & -1 & 2 \\ 0 & 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -4 \, a + b + 2 \\ a \\ b + 2 \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 41

E3 - Solving systems (ver. 41)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} - x_{3} + x_{4} &= 0 \\ x_{1} - 3 \, x_{4} &= 1 \\ 2 \, x_{1} - x_{2} - 8 \, x_{4} &= 2 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & 0 & -1 & 1 & 0 \\ 1 & 0 & 0 & -3 & 1 \\ 2 & -1 & 0 & -8 & 2 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -3 & 1 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & -1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} 3 \, a + 1 \\ -2 \, a \\ -2 \, a - 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 42

E3 - Solving systems (ver. 42)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{3} + x_{4} &= -3 \\ x_{2} + 3 \, x_{3} - 7 \, x_{4} &= 4 \\ -x_{2} - 2 \, x_{3} + 5 \, x_{4} &= -3 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 0 & -1 & 1 & -3 \\ 0 & 1 & 3 & -7 & 4 \\ 0 & -1 & -2 & 5 & -3 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & -1 & -2 \\ 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 1 & -2 & 1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a - 2 \\ a + 1 \\ 2 \, a + 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 43

E3 - Solving systems (ver. 43)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 4 \, x_{2} - 3 \, x_{3} + 7 \, x_{4} &= -7 \\ x_{2} - 2 \, x_{3} &= 0 \\ -x_{1} - 2 \, x_{2} - 6 \, x_{4} &= 6 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & 4 & -3 & 7 & -7 \\ 0 & 1 & -2 & 0 & 0 \\ -1 & -2 & 0 & -6 & 6 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 2 & -2 \\ 0 & 1 & 0 & 2 & -2 \\ 0 & 0 & 1 & 1 & -1 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \, a - 2 \\ -2 \, a - 2 \\ -a - 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 44

E3 - Solving systems (ver. 44)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} - x_{2} + 3 \, x_{3} + 3 \, x_{4} &= 2 \\ -2 \, x_{1} + x_{2} - 5 \, x_{3} - 3 \, x_{4} &= -4 \\ x_{2} - x_{3} - 3 \, x_{4} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 1 & -1 & 3 & 3 & 2 \\ -2 & 1 & -5 & -3 & -4 \\ 0 & 1 & -1 & -3 & 0 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 0 & 2 \\ 0 & 1 & -1 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \, a + 2 \\ a + 3 \, b \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 45

E3 - Solving systems (ver. 45)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + 2 \, x_{2} - 4 \, x_{3} &= 8 \\ -x_{1} - x_{2} + 2 \, x_{3} &= -3 \\ 2 \, x_{2} - 3 \, x_{3} &= 7 \\ -x_{1} + 2 \, x_{3} &= -4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 2 & -4 & 8 \\ -1 & -1 & 2 & -3 \\ 0 & 2 & -3 & 7 \\ -1 & 0 & 2 & -4 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \\ -1 \\ -3 \end{array}\right] \right\}\]


Example 46

E3 - Solving systems (ver. 46)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -2 \, x_{2} + 6 \, x_{3} - 3 \, x_{4} &= -7 \\ x_{2} - 3 \, x_{3} + 2 \, x_{4} &= 5 \\ -x_{1} - x_{2} - 2 \, x_{4} &= -7 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 0 & -2 & 6 & -3 & -7 \\ 0 & 1 & -3 & 2 & 5 \\ -1 & -1 & 0 & -2 & -7 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 3 & 0 & 2 \\ 0 & 1 & -3 & 0 & -1 \\ 0 & 0 & 0 & 1 & 3 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -3 \, a + 2 \\ 3 \, a - 1 \\ a \\ 3 \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 47

E3 - Solving systems (ver. 47)

Show how to find the solution set for the following system of linear equations.

\begin{align*} x_{1} + x_{2} - 3 \, x_{3} &= 2 \\ x_{1} + 2 \, x_{2} - 5 \, x_{3} &= 4 \\ 5 \, x_{1} + 2 \, x_{2} - 8 \, x_{3} &= 2 \\ 2 \, x_{1} + 2 \, x_{2} - 4 \, x_{3} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 1 & 1 & -3 & 2 \\ 1 & 2 & -5 & 4 \\ 5 & 2 & -8 & 2 \\ 2 & 2 & -4 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \\ -2 \\ -2 \end{array}\right] \right\}\]


Example 48

E3 - Solving systems (ver. 48)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{1} - 3 \, x_{2} - 3 \, x_{3} - 3 \, x_{4} &= 8 \\ x_{2} - x_{3} + 5 \, x_{4} &= -2 \\ x_{1} + 3 \, x_{2} + 2 \, x_{3} + 5 \, x_{4} &= -8 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} -1 & -3 & -3 & -3 & 8 \\ 0 & 1 & -1 & 5 & -2 \\ 1 & 3 & 2 & 5 & -8 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 3 & -2 \\ 0 & 0 & 1 & -2 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} -2 \\ -3 \, a - 2 \\ 2 \, a \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]


Example 49

E3 - Solving systems (ver. 49)

Show how to find the solution set for the following system of linear equations.

\begin{align*} -x_{2} - x_{3} - 2 \, x_{4} &= 3 \\ 2 \, x_{1} - x_{2} - 3 \, x_{3} - 8 \, x_{4} &= -5 \\ -x_{1} + x_{3} + 3 \, x_{4} &= 4 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrrr|r} 0 & -1 & -1 & -2 & 3 \\ 2 & -1 & -3 & -8 & -5 \\ -1 & 0 & 1 & 3 & 4 \end{array}\right] \sim \left[\begin{array}{rrrr|r} 1 & 0 & -1 & -3 & -4 \\ 0 & 1 & 1 & 2 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a + 3 \, b - 4 \\ -a - 2 \, b - 3 \\ a \\ b \end{array}\right] \middle| a,b \in \mathbb{R} \right\}\]


Example 50

E3 - Solving systems (ver. 50)

Show how to find the solution set for the following system of linear equations.

\begin{align*} 4 \, x_{1} - 8 \, x_{2} + 4 \, x_{3} &= 0 \\ 2 \, x_{1} - 3 \, x_{2} + x_{3} &= 1 \\ -x_{1} + 7 \, x_{2} - 6 \, x_{3} &= 5 \\ -x_{1} + 2 \, x_{2} - x_{3} &= 0 \\ \end{align*}

Answer.

\[\left[\begin{array}{rrr|r} 4 & -8 & 4 & 0 \\ 2 & -3 & 1 & 1 \\ -1 & 7 & -6 & 5 \\ -1 & 2 & -1 & 0 \end{array}\right] \sim \left[\begin{array}{rrr|r} 1 & 0 & -1 & 2 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

\[\left\{ \left[\begin{array}{r} a + 2 \\ a + 1 \\ a \end{array}\right] \middle| a \in \mathbb{R} \right\}\]