SOLUTION: My daughter needs serious help with Linear Programming using Graphing Calculator.  She has a problem she has been trying to solve for over a month, I have tried to help her and sea

You don't need a calculator except to check and do
basic calculations.  This is a graphing problem to
do by hand on graph paper.  

Maximize P = 13x + 2y  

subject to these constraints:

-3x + 2y ≦ 8
-8x +  y ≧ -48
x ≧ 0, y ≧ 0

[Usually P stands for profit and x and y are variables
for how many items to make or buy wholesale to sell
retail.  In practical applications there are many
variables and many inequalities.]

Those last two inequalities tell us that we don't
need any negative numbers for our graph.  So we just
need the upper right side of the graph, the first 
quadrant.

We plot the boundary lines on graph paper:

The boundary lines are the inequalities 
with = signs instead of inequality symbols.

First we draw the boundary line of the first inequality,
which is: 

-3x + 2y = 8 

We get two points, say the y-intercept (0,4) and (4,10) 
and draw a red line through them, like this:



Next we check to see whether the solutions are 
above that line or below it.

We substitute the origin (0,0) in the original inequality
to see if the origin is a solution to the inequality.  If
it is then ALL the poins on the same side of the boundary
line that the origin is on.  If it is not a solution, then
the solutions are on the side of the boundary line that the 
origin is NOT on.

So we test (0,0) by substituting x=0 and y=0 in the first
inequality to see if it is a solution.  
           -3x + 2y ≦ 8
       -3(0) + 2(0) ≦ 8
                  0 ≦ 8  That's true so the solutions are
below the red boundary line. 

--------------------------

Next we draw the boundary line of the second inequality,
which is: 

-8x + y = -48

by getting two points, the x-intercept (6,0) and (7,8).
We draw a green line through them, like this:



Next we check to see whether the solutions are 
left of right of it.

As with the first inequality, we test (0,0) by substituting
x=0 and y=0 in  the original inequality to see if it is a
solution.  -8x + y ≧ -48
       -8(0) + (0) ≧ -48
                 0 ≧ -48  That's true so all the solutions 
are on the side of the green line that the origin is on,
that is, left of the green line:



So now we know that the set of feasible solutions are the
points inside the 4 sided figure, below the red line and
left of the green line.   Next we find all the  corner 
points of that 4-sided figure.  We plotted two of them 
already, the y-intercept of the first line (0,4) and the 
x-intercept of the second line (6,0).

And we can see that (0,0) is a corner point.

We have one other corner point to find, the point where the 
red line crosses the green line, so we solve the system of
equations:

-3x + 2y =   8
-8x +  y = -48

by either the substitution or the elimination (addition) mthod,

Solve the second for y,  y=-48+8x and substitute in
       -3x + 2y = 8
-3x + 2(-48+8x) = 8
 -3x - 96 + 16x = 8
       13x - 96 = 8
            13x = 104
              x = 8
Substitute in y = -48 + 8x
              y = -48 + 8(8)
              y = -48 + 64
              y = 16

So the corner point where the red and green lines cross
is (8,16)

Now we label all four corner points:



Even though all the points in and on the feasible region are
possible solutions, the maximum and minimum points will always
be one of the corner points of the feasible region.  So we make
a table of corner points to find the value of the objective
function, the equation for P, that we are to maximize:

Corner point  Objective function       Value of P
                P = 13x + 2y
  (0,0)         P = 13(0)+2(0) = 0          0   <-- minimum
  (0,4)         P = 13(0)+2(4) = 8          8
 (8,16)         P = 13(8)+2(16) = 136     136   <-- maximum
  (6,0)         P = 13(6)+2(0) = 78        78        

Answer:
The best strategy is to choose x=8 and y=16 for a maximum
profit of $136.    

Edwin