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