Question 1207811
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x = number of tulips
y = number of roses
Each is a nonnegative integer (0,1,2,3,...)
Symbolically we would write {{{x >= 0}}} and {{{y >= 0}}}


N = x+y = total number of flowers
This is the objective function
Sarah wants the most possible flowers, so we want to make N as large as possible.


Each tulip requires 2 square feet of space.
x of those tulips require 2x square feet of space.
Each rose requires 3 square feet of space.
y of those roses require 3y square feet of space.


When considering both flowers, they require a total area of 2x+3y square feet.
The garden having a total area of 30 square feet leads to the constraint {{{2x+3y <= 30}}}


We're told that "Sarah does not want more tulips than roses"
So either the two flower counts are the same, or she wants more roses than tulips.
{{{roses >= tulips}}} which leads to {{{y >= x}}}



The constraints are
{{{system(x >= 0, y >= 0, y >= x, 2x+3y <= 30)}}}


The first two inequalities will mean we focus only on the upper right quadrant (known as quadrant 1).
Graph the line y = x. Shade above it to represent {{{y >= x}}}
Graph the line 2x+3y=30. This line goes through (0,10) and (15,0). Shade below it to represent {{{2x+3y <= 30}}}


Here is what it looks like when we graph the last two inequalities on the same xy grid. 
*[illustration UploadedScreenshot_55.png]
It's a bit of a mess isn't it? 


I used Desmos to graph.


Desmos Link
<a href="https://www.desmos.com/calculator/jfra5nmlja">https://www.desmos.com/calculator/jfra5nmlja</a>


The two regions overlap to form this triangular region. 
Again only focus on the upper right quadrant.
*[illustration UploadedScreenshot_56.png]
We have a much cleaner graph.


Desmos Link
<a href="https://www.desmos.com/calculator/5lkfm5urii">https://www.desmos.com/calculator/5lkfm5urii</a>


The vertices of the triangle are
(0,0)
(0,10)
(6,6)
Each vertex is found by intersecting the boundary lines.
As an example, we solve the system {2x+3y=30,y=x} to determine the vertex (x,y)=(6,6)


Plug each vertex into the objective function
Let's plug in (0,0)
N = x+y
N = 0+0
N = 0
Let's plug in (0,10)
N = x+y
N = 0+10
N = 10
Let's plug in (6,6)
N = x+y
N = 6+6
N = 12


Sarah will get the most flowers (12 total) when she plants x = 6 tulips and y = 6 roses.
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