1. Describe the linear programming situation for this system of inequalities.

x<(or equal too)1 y>(or equal too)0
3x + y<(or equal too)5

2. Describe the linear programming situation for this system of inequalities where you are asked to find the maximum value of f(x, y) = x + y.

x>(or equal too) 0 y>(or equal too)0
6x + 3y<(or equal too)18 x + 3y<(or equal too)9

left (or at) of x = 1

above or on x axis
below or on y = -3x+5
which hits x axis at x = 5/3
and hits the y axis at y = 5
sketch that
You see that the vertical line x = 1 hits our sloped line somewhere above the x axis
find that point
x = 1
y = -3(1) + 5 = 2
so
we have a corner at (1, 2)
everything in the upper half plane left of the sloping line and the vertical line x = 1

It did not say x had to be positive, so it goes left forever. That would not be likely in a real linear programming problem.

2. Describe the linear programming situation for this system of inequalities where you are asked to find the maximum value of f(x, y) = x + y.

x>(or equal too) 0 y>(or equal too)0
6x + 3y<(or equal too)18 x + 3y<(or equal too)9
=================================
In first quadrant due to x>/=0 and y>/= 0

6x + 3y<(or equal too)18
is
y </= -2x + 6
x and y axis intercepts at x = 3 and at y = 6
below that line

3 y </= -x + 9
y </= -(1/3) x + 3
intercepts at x = 9 and at y = 3
below that line

we need the corner where those sloped lines hit
-(1/3) x + 3 = -2x+6
-x + 9 = -6 x + 18
5 x = 9
x = 9/5
then y = -(1/3)(9/5) + 3
= 3 - 3/5 = 12/5

So we have a corner at (9/5 , 12/5)

so three corners to test
(0 , 3) , (9/5 , 12/5) , (3 , 0)
at (0,3) x+y = 3
at (9/5,12/5) x+y = 21/5 = 4. something
at (3,0) x+y = 3 again
so the max is at the intersection of the sloped lines and is 21/5

1. To describe the linear programming situation for this system of inequalities, we first need to understand what linear programming is. Linear programming is a mathematical method used to optimize (maximize or minimize) a linear objective function subject to a set of linear constraints. In this situation, we have three inequalities:

a) x ≤ 1
b) y ≥ 0
c) 3x + y ≤ 5

Inequality (a) represents the constraint that x should be less than or equal to 1. Inequality (b) represents the constraint that y should be greater than or equal to 0. Inequality (c) represents the constraint that the linear combination of 3 times x and y should be less than or equal to 5.

So, the linear programming situation for this system of inequalities is to optimize a linear objective function (which is not provided) subject to these constraints.

2. To describe the linear programming situation for this system of inequalities, we need to understand the objective function and the constraints. Here, the objective function is f(x, y) = x + y, which we are asked to maximize.

The system of inequalities consists of four constraints:

a) x ≥ 0
b) y ≥ 0
c) 6x + 3y ≤ 18
d) x + 3y ≤ 9

Inequality (a) states that x should be greater than or equal to 0, indicating a non-negative value for x. Inequality (b) states that y should also be greater than or equal to 0, indicating a non-negative value for y. The inequality (c) indicates that the linear combination of 6 times x and 3 times y should be less than or equal to 18, setting a limit on the sum of x and y. Lastly, the inequality (d) states that the linear combination of x and 3 times y should be less than or equal to 9, providing another constraint on the sum of x and y.

In this linear programming situation, we are tasked with maximizing the objective function f(x, y) = x + y while adhering to the given constraints.