Solving Linear Programming Problems Graphically

To solve the linear programming problem
  subject to the constraints
\[2a+b \geq 12\]

\[a+b \geq 9\]

\[a+3b \geq 15\]

and of course  
\[a, \: b \geq 0\]
We plot these inequalities - as equalities - on a graph. We seek to maximise the objective function, so draw the line  
  for various values of  
, seeking to find the maximum value of  
  for which the objective function is in or on a boundary of the feasible region (the part of the graph that satisfies all the constraints.

This is at the intersection of the lines  
Solving the equations


\[a=4.2, \: b=3.6\]

The value of the objective is
\[30a+40b=30 \times 4.2+40 \times 3.6=270\]