Simultaneous Linear Equations/Examples
Examples of Simultaneous Linear Equations
Arbitrary System $1$
The system of simultaneous linear equations:
| \(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 2 x_1 - x_2 + x_3\) | \(=\) | \(\ds 2\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds 4 x_1 + x_2 - x_3\) | \(=\) | \(\ds 1\) |
has as its solution set:
| \(\ds x_1\) | \(=\) | \(\ds -\dfrac 1 2\) | ||||||||||||
| \(\ds x_2\) | \(=\) | \(\ds \dfrac 1 2\) | ||||||||||||
| \(\ds x_3\) | \(=\) | \(\ds \dfrac 3 2\) |
Arbitrary System $2$
The system of simultaneous linear equations:
| \(\text {(1)}: \quad\) | \(\ds x_1 + x_2\) | \(=\) | \(\ds 2\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 2 x_1 + 2 x_2\) | \(=\) | \(\ds 3\) |
has no solutions.
Arbitrary System $3$
The system of simultaneous linear equations:
| \(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 2 x_1 - x_2 + x_3\) | \(=\) | \(\ds 2\) |
has as its solution set:
| \(\ds x_1\) | \(=\) | \(\ds 1 - \dfrac t 3\) | ||||||||||||
| \(\ds x_2\) | \(=\) | \(\ds \dfrac t 3\) | ||||||||||||
| \(\ds x_3\) | \(=\) | \(\ds t\) |
where $t$ is any number.
Arbitrary System $4$
Simultaneous Linear Equations/Examples/Arbitrary System 4
Arbitrary System $5$
Simultaneous Linear Equations/Examples/Arbitrary System 5
Arbitrary System $6$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + y + 2 z\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds -x + z\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds -x + y + 4 z\) | \(=\) | \(\ds -3\) |
$S$ has as its solution set:
| \(\ds x\) | \(=\) | \(\ds z + 1\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds z - 2\) |
where $z$ can be any number.
Arbitrary System $7$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + 2 y + z\) | \(=\) | \(\ds 2\) | ||||||||||||
| \(\ds -x + 2 y\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds 5 x - 2 y + 2 z\) | \(=\) | \(\ds 7\) |
$S$ has as its solution set:
| \(\ds 2 x + z\) | \(=\) | \(\ds 3\) | ||||||||||||
| \(\ds 4 y + z\) | \(=\) | \(\ds 1\) |
where $z$ can be any number.
Arbitrary System $8$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x - y - z\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds 2 x - y\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds 2 x + 2 z\) | \(=\) | \(\ds 1\) |
$S$ is inconsistent and so has no solutions.
Arbitrary System $9$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + 2 y + z\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds x + y + z\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds -x + z\) | \(=\) | \(\ds 1\) |
$S$ has the single solution:
| \(\ds x\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds z\) | \(=\) | \(\ds 1\) |
Arbitrary System $10$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + 2 y - z + w\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds 2 x + y + z\) | \(=\) | \(\ds 2\) |
$S$ has as its solution set:
| \(\ds 3 x\) | \(=\) | \(\ds 3 - z + w\) | ||||||||||||
| \(\ds 3 y\) | \(=\) | \(\ds 3 z - 2 w\) |
where $z$ and $w$ can be any numbers.
Arbitrary System $11$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + y - z\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds y + z\) | \(=\) | \(\ds 2\) | ||||||||||||
| \(\ds x + 2 z\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds x - y + 5 z\) | \(=\) | \(\ds 1\) |
$S$ has the single solution:
| \(\ds x\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds \dfrac 3 2\) | ||||||||||||
| \(\ds z\) | \(=\) | \(\ds \dfrac 1 2\) |
Arbitrary System $12$
Let $S$ denote the system of simultaneous linear equations:
| \(\ds x + 2 y + 3 z + w\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds -x + y + x - w\) | \(=\) | \(\ds 2\) | ||||||||||||
| \(\ds x + 5 y + 7 z + w\) | \(=\) | \(\ds 1\) |
$S$ is inconsistent and so has no solutions.