Linear Equations

Definitions

consistent system : has a unique solution or infinitely many solutions

inconsistent system : has no solutions at all

$y=mx+b$ is slope-intercept form. $Ax+By=C$ is the form we’ll mostly deal w/ in Eastern Oregon University (EOU)‘s Linear Algebra class.

There can be multiple variables. $a_1{x}_1+a_2{x}_2+...a_n{x}_n=b$ where $a$ is a coefficient and $x$ is the variable to create a line.

A “system of linear equations” usually means more than 1 line (though not always), like:

$$\begin{array}{r}{x}_1+3x_2+5x_3=-2\\ \\ 3x_1+7x_2+7x_3=6\\ \\ x_2+4x_3=-5\end{array}$$

There may not be a solution to where to find an intersection. System could intersect (one solution), run parallel (no solution) or be the exact same line (infinitely many solutions).

We could record the coefficients above in a coeffient matrix.

\begin{matrix} 1 & 3 & 5 \\ 3 & 7 & 7 \\ 0 & 1 & 4 \end{matrix}

or an augmented matrix (same as above, but records what it results):

\begin{array}{ccc|c} 1 & 3 & 5 & -2 \\ 3 & 7 & 7 & 6 \\ 0 & 1 & 4 & -5 \end{array}