ok here we are going to maximize the function C equals x plus y given the
constraints that why has to be greater than or equal to 0 X has to be greater
than or equal to 0 4x plus 2y must be less than or equal to 8 and 2x
minus y has to be less than or equal to 0. Notice the first two
constraints y greater than or equal to 0 and X greater than or equal to 0 that
simply means that we're going to be our are feasible region is going to be in
the first quadrant quadrant one so we'll keep that in mind and then i'm going to
start graphing the inequalities 4x + 2 y less than or equal to 8 and 2 X minus y
less than or equal to 0 so I'm going to start off by making both of these into
equations so those these are just going to be lines and the graph these i'm
going to find the x and y-intercepts of each so to find the x-intercept we can
simply let y equal zero for the first equation will be left with 4x equals
eight if we divide both sides by 4 x equals 2 for the second one well we'll
have 2x equals 0 if we divide both sides by 2 will just get x equals 0 so this
one's a not going to be as useful because well we'll see in just a second
to find the y-intercept we let X equal 0 and if we let X equal 0 we would have 2y
equals 8
if we divide both sides by 2 will get y equals four
well ok back to what i was saying a second ago if we let X equal 0 will have
negative y equals 0 which tells us y equals 0 so here we know we're
getting a point 2 comma 0 and 0 comma 4 on our line but for our second line
we're only getting the point 0 0 and again we're getting 00 so what we can do
though
2x minus y equals 0 we could always write that as 2 x equals y
then we can graph it just using slope intercept form
ok so let's start graphing here again we said we're in the first quadrant so I'm
going to make my my graph so that we just really only see the first quadrant
ok so the first equation for x plus 2y equals 8 there's a point 2 comma 0 and
also at 0 comma 4 against it was less than or equal to the original constraint
we can use solid to solid line and then likewise we have y equals 2x so that's
going to have a y-intercept of 0 and then we go up to over 1 up to over 1
etcetera
ok so now we simply have to determine the feasible region so what I'm going to
do is use a test . again keeping in mind that this has to be in the first
quadrant so 4x plus 2y has to be less than or equal to 8 and our other
our other inequality was 2x minus y less than or equal to 0 so for my line for x
plus 2y equals 8 to test the inequality i'm going to use a point not on that
line and i met again just going to use the origin 0 0 so again the other line
does go through 00 but you can almost forget about that line for a second so
as long as I take a point off of the line 4x plus 2y equals 8
I can use that as a test point so for my first inequality i'm going to test the
point 0 0 and certainly if you plug in 0 for X and 0 for y we would get that 0 is
less than or equal
8 so I know for my first inequality i would have to shade below for the second
12 X minus y less than or equal to 0
well now we can't use the test point 00 so we have to use some other point I don't
know how about the point 1 comma 0 so I'm going to test the point 1 comma 0 so
if we plug 1 in for X and 0 in for why i have to ask myself does that satisfy our
inequality
well we'll have to say is 2 less than or equal to 0 certainly not so for my
second inequality the test point was below it which means we need to shade above it
and again since we're trapped in the first quadrant we know that our
feasible region is going to be this little triangular region it's a touching
the the y-axis and it's easy to see that this is going to cross at the point 1
comma 2 so that's the benefit of having a nice good graph
ok so we can even label all the corner points we've got 1 comma 2 we've got
zero comma 0 the origin and then we also have the point 0 comma 4 so now we
simply need to take these three points put them into the function that we were
trying to maximize and determine which of those gives us our largest value so
we've got C equals we want to maximize c equals x plus y and again we've got the
points (0,0) (0,4) and the other one was 1 comma 2 well the arithmetic here is
pretty easy i think so clearly if you plug in 0 0 will get that c equals 0
0 plus 0 we put in 0 comma 4 will get that c equals 0 + 4 or we will get the value of
four
and if we plug in 1 comma 2 into our function will get 1 plus 2 way that only
gives us three so it says the maximum value possible will be the value of
positive 4 and again that's going to occur at the point 0 comma 4