System of Linear Equations

A system of linear equations a set of relationships of variables which must all be satisfied simultaneously. So while each equation itself may have an infinite number of solutions, there may only be one or no solution when solved in tandem with another. Also, the relations must be linear, thus the name "linear equations".

Here is the notation for a linear equation of n variables and m equations

$ \begin{align} a_{1,1}x_1&+&a_{1,2}x_2&+&\cdots&+&a_{1,n}x_n &=&a_{1,n+1}\\ a_{2,1}x_1&+&a_{2,2}x_2&+&\cdots&+&a_{2,n}x_n &=&a_{2,n+1}\\ \vdots & & \vdots\ &&\vdots&&\vdots &&\vdots\\ a_{m,1}x_1&+&a_{m,2}x_2&+&\cdots&+&a_{m,n}x_n &=&a_{m,n+1}\\ \end{align} $

Where all a's are scalars and the ones on the right side of the equations are constants while the rest are coefficients for the variables.

Systems of equations can be solved in several ways. They can be solved using Gauss-Jordan elimination, consecutive substitution, or for systems with only two variables you can use graphical methods by representing each equation as a line and finding the intersections.

One of the most common things to do is to represent the system as a matrix multiplying a vector of variables. Using the above example we'd get an mxn matrix:

$ \begin{bmatrix} a_{1,1}&a_{1,2}&\cdots&a_{1,n}\\ a_{2,1}&a_{2,2}&\cdots&a_{2,n}\\ \vdots & \vdots\ &\vdots&\vdots\\ a_{m,1}&a_{m,2}&\cdots&a_{m,n} \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n\end{bmatrix} = \begin{bmatrix} a_{1,n+1}\\ a_{2,n+1}\\ \vdots\\ a_{m,n+1}\end{bmatrix} $

You can then solve by using the various methods for solving a matrix multiplication.