What are the possible solutions to systems of linear equations

2 Equations, 2 variables

• So two lines.

$$\begin{array}{r}{a}_{11}{x}_1+a_{12}{x}_2=b_1\\ a_{21}{x}_1+a_{22}{x}_2=b_2\end{array}$$

• Three possibilities
  • No intersection
  • Always intersecting (same line basically).
  • One intersection.

3 Equations, 3 variables

• 3 planes.