Systems of Linear Equations

Definition: A System of Linear Equations is a collection of $m$ linear equations on $n$ variables represented as: $a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1 \\a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2 \\ \vdots \quad \quad \quad \vdots \quad \quad \: \: \quad \vdots \quad \quad \vdots \: \: \: \\ a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m$.

For example, the following is a system of 3 linear equations of the 4 variables $w, x, y, z$:

(1)

\begin{align} 3w + 2x - 4y + 4z = 5 \\ 2w - 4x + 5y - 6z = 2 \\ -w + 3x - 0y + 2z = 12 \end{align}

Definition: A Solution to a linear equation of $n$ variables $(x_1, x_2, ..., x_n) = (s_1, s_2, ..., s_n)$ is an ordered $n$-tuple that satisfies all equations in that linear system. A system of linear equations that has 1 or infinitely many solutions is said to be Consistent, while a system that has no solutions is said to be Inconsistent.

For example, consider the following system of 2 linear equations of 2 variables containing the linear equations $x - y = -2$ and $x + y = 4$, which represents two lines. A solution to this system exists at the intersection of the two lines, and thus, the coordinates $(1, 3)$ is a solution. We can verify this solution by plugging these values into both equations.

(2)

\begin{align} x - y = ? \quad \quad \quad x + y = ? \\ 1 - 3 = -2 \quad \quad \quad 1 + 3 = 4 \end{align}

The following image illustrates the graphs $x - y = -2$ and $x + y = 4$, where we can see that the point $(1, 3)$ is the intersection/solution to this system.

Some times, we may have infinitely many solutions to a linear system. For example, consider the system of equations from the equations $x + y + z = 1$ and $x + y + z - 1 = 0$. Both of these equations represent the same plane, and hence, these planes coincide with each other. So every set of values $(x, y, z)$ such that $x + y + z = 1$ instantly satisfy the other equation $x + y + z - 1 = 0$, and so there are infinitely many solutions to this system.

The last possibility we can have regarding a system of linear equations is having no solutions. For example, consider a system of two parallel lines. For example, the system containing only the equations $x + y = 4$ and $x + y = 6$ illustrated below:

In this case, there are no solutions as these lines will never intersect one another.