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How do matrices represent the coefficients in simultaneous equations?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, a system of simultaneous linear equations is represented by a matrix by organizing the coefficients of the unknowns into a rectangular array.12

Specifically, for a system of $m$ linear equations with $n$ unknowns, the coefficients are arranged into an $m \times n$ matrix, often denoted as $A$ . When this system is expressed as the matrix equation $A X = B$ :2

$A$ is the matrix containing the coefficients of the unknowns.2
$X$ is a column vector (an $n \times 1$ matrix) representing the unknown variables.2
$B$ is a column vector containing the constant terms from the right-hand side of the equations.2

To include the constant terms directly within the coefficient matrix for calculation purposes, mathematicians use an augmented matrix, which adds the constant terms as an extra column to the coefficient array. This representation allows for the use of efficient matrix techniques to solve systems that would be cumbersome to handle through simple substitution.31