Question 989537
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There are six different solutions if you allow the case where one or more of the animal types are not represented at all, 5 valid solutions otherwise.


Let *[tex \Large x] represent the number of cats, *[tex \Large y] represent the number of dogs, *[tex \Large z] represent the number of fish.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ y\ =\ 100\ -\ z]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5x\ +\ 10y\ =\ 1000\ -\ 100z]


Solve this by cases, where *[tex \Large z] takes on values in the range 0 through 10.  If you wish to disallow the case where any animal type is not represented at all, then start at 1.  For each value of z you choose, adjust the value of the constant term and then solve the 2X2 system for *[tex \Large x] and *[tex \Large y].  At some point, you will start getting a negative number for the *[tex \Large y] variable, so you can discard that value of *[tex \Large z] and anything larger.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \