Question 536569
We want to plot the boundaries of the feasibility region, and find their intersection points.
Obviously x and y are defined so that they cannot be negative, because two of the boundaries are the x- and y-axes (y=0 and x=0).
Another boundary is the horizontal line y=45 (y cannot exceed 45).
{{{5x+6y=420}}} is a line (in blue in the drawing below) that intersects the x- and y-axes at the points where
{{{5x=420}}} --> {{{x=84}}} and
{{{6y=420}}} --> {{{y=70}}}
Where it intersects y=45,
{{{5x+6*45=420}}}-->{{{x=30}}}
The feasibility region can be represented as the trapezoid OABC below.
{{{drawing( 330, 300, -10, 100, -10, 90,
  grid(10),
  locate( 1, 5, O ), locate( 1, 50, A ),
  locate( 31, 50, B ), locate( 84, 5, C ),
  blue(line( 0, 70, 84, 0 )),
  green(line( 0, 45, 90, 45 ))
)}}} O=(0,0) A=(0,45) B=(30,45) C=(84,0)
For each value of P, P= 6x + 5y will be a straight line. Different values of P will produce parallel lines. As you increase P, the line will move until it reaches a boundary point or a boundary line. It is obvious that point O gives P=0, and that point B will give a greater P than point A. If you calculate P for the coordinates of points B and C, you will find the maximum. I can see that it will be at a point, and not along the blue line, because the P= 6x + 5y lines have a different (steeper) slope than the blue line. In fact, I can even see where the maximum will be, without calculating. Can you?