Draw a phase plane for the following system of ODEs.

\[x'= -2 \, x - 3 \, y + 3 \]

\[y'= x - 5 \, y - 8 \]

Be sure to include and label the following features in your sketch:

The isoclines \(x'=0\) and \(y'=0\).
Arrow systems describing trajectories of solutions.
The equillibrium of the system.

Answer:

The isoclines cross at the equillibrium \( \left(3, -1\right) \), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

The right region has a northwest arrow system, the top region has a southwest arrow system, the left region has a southeast arrow system, and the bottom region has a northeast arrow system.