Question 1143452
A cat breeder has the following amounts of cat food: 120 units of tuna, 110 units of liver, and 70 units of chicken. 
To raise a Siamese cat, the breeder must use 2 units of tuna, 2 of liver and 3 of chicken, while raising a Persian cat requires 
3, 2, and 1 units respectively per day. 
If a Siamese cat sells for $14 and a Persian cat sells for $16, how many of each 
should be raised in order to obtain maximum gross income? 
Set up the system of equations and initial simplex tableau. 
<pre>
Let x = no. of Siamese cats to be raised
Let y = no. of Persian cats to be raised
We make the tuna inequality to make sure we don't run out of tuna,
{{{2x + 3y <= 120}}}
We make the liver inequality to make sure we don't run out of liver.
{{{2x + 2y <= 110}}}
We make the chicken inequality to make sure we don't run out of chicken.
{{{3x + 1y <= 70}}}
We make the objective equation for the income that we want to maximize.
P = 14x + 16y
So We add NOB-NEGATIVE slack variables to take up the slack and turn the 3
inequalities into equations:
{{{2x + 3y + s[1] = 120}}}
{{{2x + 2y + s[2] = 110}}}
{{{3x + 1y + s[3]  = 70}}}
Then we rearrange the objective equation so that the variable terms are
negative and we have 0 on the right:
{{{-14x - 16y + P = 0}}}
So we have this system of equations:
{{{system(
matrix(4,11,
  2x,""+"", 3y,""+"",s[1],"  ","  ","  ","  ",""="",120,
  2x,""+"", 2y,""+"","  ",s[2],"  ","  ","  ",""="",110,
  2x,""+"", 3y,""+"","  ","  ",s[3],"  ","  ",""="", 70,
-14x,""-"",16y,""+"","  ","  ","  ",""[""],P, ""="",  0))}}}

We make the augmented matrix for the system:

{{{(matrix(6,9,

x,y,s[1],s[2],s[3],"|",P,"|",k,
2,3,   1,   0,   0,"|",0,"|",120,
2,2,   0,   1,   0,"|",0,"|",110,
2,3,   0,   0,   1,"|",0,"|",70,
""-"",""-"",""-"",""-"",""-"",""-"",""-"",""-"",""-"",
-14,-16,0,0,0,"|",1,"|",0))}}}


I'm going to stop here.  If you need help finishing,
tell me so in the form below, and I'll get back to
you by email. No charge.  I'm an 82-year-old retired
math professor, and I just do this for fun. lol

Edwin</pre>