# 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 $6$

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.