Solving Linear Systems

Unique Solution. Finding the solution set of a system of two linear equations in two variables is easy because it amounts to finding the intersection of two lines. (a) A typical problem is

$\begin{displaymath} \left\{ \begin{array}{l} x_1-2x_2 =-1 \\ -x_1+3x_2 =3 \end{array} \right. \end{displaymath}$

The graphs of these equations are lines, which we denote by $L_1$ and $L_2$ . A pair of numbers $(x_1,x_2)$ satisfies both equations in the system if and only if the point lies on both $L_1$ and $L_2$ . In the system above, the solution is the single point $(3,2)$ .

General Solutions. Two lines need not intersect in a single point. (b) They could be parallel as in the following system:

$\begin{displaymath} \left\{ \begin{array}{l} x_1-2x_2 =-1 \\ -x_1+2x_2 =3 \end{array} \right. \end{displaymath}$

Or, (c) they could coincide and hence intersect at every point on the line as in the following system:

$\begin{displaymath} \left\{ \begin{array}{l} x_1-2x_2 =-1 \\ -x_1+2x_2 =1 \end{array} \right. \end{displaymath}$

A system of linear equations has (a) exactly one unique solution, or (b) no solution, or (c) infinitely many solutions.