math111_logo System of Linear Equations


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.

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

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