System of linear equations

A system of linear equations is a collection of linear equations that are used to solve for the values of multiple variables. These equations can be represe...

A system of linear equations is a collection of linear equations that are used to solve for the values of multiple variables. These equations can be represented in various forms, including:

Matrix form:

$\begin{bmatrix}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{bmatrix}\begin{bmatrix}x_1 \\\ x_2 \\\ x_3\end{bmatrix} = \begin{bmatrix}b_1 \\\ b_2 \\\ b_3\end{bmatrix}$

Vector form:

$\mathbf{x} = \begin{bmatrix}x_1 \\\ x_2 \\\ x_3\end{bmatrix}, \mathbf{a} = \begin{bmatrix}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{bmatrix}, \mathbf{b} = \begin{bmatrix}b_1 \\\ b_2 \\\ b_3\end{bmatrix}$

General form:

$ax + by + cz = d$

where a, b, and c are constants and d is a constant.

The solution to a system of linear equations is a set of values for the variables that makes all of the equations true.

Solving a system of linear equations can be done using various methods, including substitution, elimination, and matrix methods. The chosen method depends on the specific system of linear equations and the available tools and techniques.

Here are some examples of linear equations:

2x + 4y = 10
x - y = 2
3x + y = 5

Solving this system of linear equations, we get the following solution:

x = 2
y = 4

These values are the only values that make all of the equations true