System of Linear Equations
System of Linear EquationsA linear equation in variables x1, x2, ..., xn is of the form
a1x1 + a2x2 + ... + anxn = b,
where the coefficients a1, a2, ..., an and the right side b are numbers. A system of linear equations is a collection of linear equations involving the same variables.
| a11x1 | + a12x2 | + ... + a1nxn | = | b1 |
| a21x1 | + a22x2 | + ... + a2nxn | = | b2 |
| ... | ... | ... | ||
| am1x1 | + am2x2 | + ... + amnxn | = | bm |
Example The following 3 equations are linear.
2x1 + 3x2 - x3 = 5,
x1 - 2x2 + 5x3 + 10x4 - 4x5 = 12,
2x + 3y - z = 1.
The equations
x1 + x2 = x3 + x4,
2(x1 - 3) = 2x3 + 3(x1 + 1) - 10x2,
u + v = 2(u - 1) - 3v + 5,
are also linear because they can be rewritten as
x1 + x2 - x3 - x4 = 0,
- x1 + 10x2 - 2x3 = 9,
- u + 4v = 3.
The equations
x1x2 = x1 + x2,
2 sinx1 + 3 cosx2 - sinx3 = 4,
x3 + x2y - xy2 - y3 = 1,
are not linear.
Example The following is a system of 3 linear equations in 3 variables.
| x1 | - x2 | + x3 | = | 2 |
| 3x1 | + x2 | - x3 | = | 2 |
| 2x1 | + 2x2 | + x3 | = | 6 |
The following is a system of 4 linear equations in 5 variables.
| x1 | +3 x2 | + 2x3 | + x5 | = | 0 | |
| - x1 | - x2 | - x3 | + x4 | = | 1 | |
| 4x2 | + 2x3 | + 4x4 | + 3x5 | = | 3 | |
| x1 | + 3x2 | + 2x3 | - 2x4 | = | 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, 3x = 2 is a system of 1 equation in 1 variable.
The following system is not linear.
| x1 | + x2 | + x3 | = | 1 |
| x12 | + x22 | + x32 | = | 1 |
The following is also not linear.
| xy | + yz | + zx | = | 0 |
| x1/2 | - y1/2 | + z1/2 | = | 2 |