What are system of linear equations?

Answer 1

Systems of linear equations are simply two or more linear equations related to each other, whether it be through having similar variables or describing similar processes.

Linear equations are equations whose only terms are constants and/or single variables raised to the first power. More than one variable is allowed in a linear equation, but it is not allowed to be multiplied with another variable. Constants are allowed to be multiplied to variables in linear equations. These equations are called "linear" due to the fact that their solution set forms a line when represented in classic Euclidean space, e.g. when graphed on the mutually perpendicular x, y, and z axes of the Cartesian coordinate system.

Some examples of linear equations are as follows:

x+2 = 3

x+y+z = 4

2*a+3*b+4*c+5*d = 10*e

If these three linear equations were all necessary to describe something mathematically, they would be a system of linear equations.

To solve a system of linear equations for every unknown, the number of variables must be less than or equal to the number of equations.

To actually solve systems of linear equations, the easiest way is to use a calculator and linear algebra.

If you had something like this:

{ 2*x+3*y = 5

{ 4*x+10*y = 7

What you would do is this:

1. Put the coefficients in a matrix:

[2 3]

[4 10]

2. Now put the answers in a matrix:

[5]

[7]

3. Invert the matrix of the coefficients and multiply that by the answer:

[2 3] ^(-1) = [1.25 -.375]

[4 10] [-.5 .25]

4. Multiply this by the matrix

[5]

[7]

[1.25 -.375] * [5]

[-.5 .25] [7]

=

[3.625]

[-0.75]

Meaning your x = 3.625 and your y = -.75