A linear equation in variables `x`_{1}`, x`_{2},
..., `x _{n}` is of the form

`a`_{1}`x`_{1} + `a`_{2}`x`_{2}
+ ... + `a _{n}x_{n}` =

where the coefficients `a`_{1}`, a`_{2},
..., `a _{n}` and the right side

a_{11}x_{1} |
+ a_{12}x_{2} |
+ ... + a_{1n}x_{n} |
= | b_{1} |

a_{21}x_{1} |
+ a_{22}x_{2} |
+ ... + a_{2n}x_{n} |
= | b_{2} |

... | ... | ... | ||

a_{m1}x_{1} |
+ a_{m2}x_{2} |
+ ... + a_{mn}x_{n} |
= | b_{m} |

Example The following 3 equations are linear.

2`x`_{1} + 3`x`_{2} - `x`_{3} = 5,

`x`_{1} - 2`x`_{2} + 5`x`_{3} + 10`x`_{4}
- 4`x`_{5} = 12,

2`x` + 3`y` - `z` = 1.

The equations

`x`_{1} + `x`_{2} = `x`_{3} + `x`_{4},

2(`x`_{1} - 3) = 2`x`_{3} + 3(`x`_{1} + 1) - 10`x`_{2},

`u` + `v` = 2(`u` - 1) - 3`v` + 5,

are also linear because they can be rewritten as

`x`_{1} + `x`_{2} - `x`_{3} - `x`_{4} = 0,

- `x`_{1} + 10`x`_{2} - 2`x`_{3} = 9,

- `u` + 4`v` = 3.

The equations

`x`_{1}`x`_{2} = `x`_{1} + `x`_{2},

2 sin`x`_{1} + 3 cos`x`_{2} - sin`x`_{3} = 4,

`x`^{3} + `x`^{2}`y` - `xy`^{2} - `y`^{3} = 1,

are not linear.

Example The following is a system of 3 linear equations in 3 variables.

x_{1} |
- x_{2} |
+ x_{3} |
= | 2 |

3x_{1} |
+ x_{2} |
- x_{3} |
= | 2 |

2x_{1} |
+ 2x_{2} |
+ x_{3} |
= | 6 |

The following is a system of 4 linear equations in 5 variables.

x_{1} |
+3 x_{2} |
+ 2x_{3} |
+ x_{5} |
= | 0 | |

- x_{1} |
- x_{2} |
- x_{3} |
+ x_{4} |
= | 1 | |

4x_{2} |
+ 2x_{3} |
+ 4x_{4} |
+ 3x_{5} |
= | 3 | |

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
- 2x_{4} |
= | 0 |

Here is a system of 3 linear equations in 1 variable.

u |
= | 2 |

3u |
= | 2 |

2u |
= | 6 |

Here is a system of 3 linear equations in 3 variables.

u |
= | 2 |

3v |
= | 2 |

2w |
= | 6 |

Finally, 3`x` = 2 is a system of 1 equation in 1 variable.

The following system is not linear.

x_{1} |
+ x_{2} |
+ x_{3} |
= | 1 |

x_{1}^{2} |
+ x_{2}^{2} |
+ x_{3}^{2} |
= | 1 |

The following is also not linear.

xy |
+ yz |
+ zx |
= | 0 |

x^{1/2} |
- y^{1/2} |
+ z^{1/2} |
= | 2 |

[next topic]