Question 797445
{{{x+y<=6}}} includes the line {{{x+y=6}}} and all the point to one side of that line.
What side?
Obviously the side with the origin (0,0), because {{{system{{{x=0,y=0}}} is a solution.
The line {{{x+y=6}}} obviously passes through the points (6,0) and (0,6) because
{{{x=0}}}-->{{{0+y=6}}}-->{{{y=6}}} and
{{{y=0}}}-->{{{x+0=6}}}-->{{{x=6}}}
 
{{{2x+y<=10}}} includes the line {{{2x+y=10}}} and all the point to one side o fthat line.
What side?
Obviously the side with the origin (0,0), because {{{system(x=0,y=0)}}} is a solution.
The line {{{2x+y=10}}} obviously passes through the points (5,0) and (0,10) because
{{{x=0}}}-->{{{2*0+y=10}}}-->{{{y=10}}} and
{{{y=0}}}-->{{{2x+0=10}}}-->{{{2x=10}}}-->{{{x=10/2}}}-->{{{x=5}}}
 
{{{x>=0}}} includes the line {{{x=0}}} (the y-axis) and all the points to the right of it, like (1,0), with {{{system(x=1,y=0)}}} that satisfy {{{x>0}}}.
 
{{{y>=0}}} includes the line {{{y=0}}} (the x-axis) and all the points above it, like (0,1), with {{{system(x=0,y=1)}}} that satisfy {{{y>0}}}.
 
The space that satisfies all those constraints is the quadrilateral OABC below.
{{{drawing(300,360,-2,8,-1,11,
grid(0),line(-2,8,8,-2),
line(-1,12,6,-2),locate(0.1,0.6,O),
locate(0.1,5.8,A),locate(3.7,2.1,B),
locate(4.5,0.6,C)
)}}} The maximum value for P is at a vertex or along one entire edge of that quadrilateral.
Point B is the solution to {{{system(2x+y=10,x+y=6)}}}, which is obviously {{{system(x=4,y=2)}}}.
 
The value of {{{P=4x+y}}}
at O(0,0) is {{{P=4*0+0=0}}};
at A(0,6) is {{{P=4*0+6=6}}};
at B(4,2) is {{{P=4*4+2=16+2=18}}}; and
at C(5,0) is {{{P=4*10+0=20+0=20}}}
So the maximum is {{{P=highlight(20)}}} at C(5,0).