Number of integral solutions of...

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mathdaemon

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Hello

x+2y+3z=30
What is the no. of integral solutions of this equation?

I know how to do the x+y+z=30(Using stars and bars) but this is new to me.

Please help
 
mathdaemon said:
Hello

x+2y+3z=30
What is the no. of integral solutions of this equation?

I know how to do the x+y+z=30(Using stars and bars) but this is new to me.

Please help
Click to expand...
I do not see an easy way to do this. If z=10, then only x=y=0, z=10 works.
If z=9 that leaves 3 left over. So y can be 0 or 1 (with the correct value of x). So 2 results.
If z=8 that leaves 6 left over. So y can be 0, 1, 2 or 3. So 4 more results.
If z=7 that leaves 9 left over. So y can be 0,1,2,3 or 4. Add on 5 more results.
...
If z=1 that leaves 27 left over. So y can be 0,1,2...13. Add on 14 results.
If z=0 that leaves 30 left over. So y can be 0,1,2,...15. !6 more results.
 
pka said:
What exactly do you by an integer solution?

\(\displaystyle (-12,~15,~4)\) is a solution.
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Why would I think (ok, bad choice of a word) that x,y and z had to be positive?
 
Jomo said:
Why would I think (ok, bad choice of a word) that x,y and z had to be positive?
Click to expand...

@Jomo, Why did you think that I was addressing you? I was not.

Who did quote at the top of the posting? It had nothing to do with you.

I would have quoted your post if you were involved.
 
mathdaemon said:
Hello

x+2y+3z=30
What is the no. of integral solutions of this equation?

I know how to do the x+y+z=30(Using stars and bars) but this is new to me.

Please help
Click to expand...

Assuming that you meant integer solutions, since
a x + b y + c z + d = 0
is the general equation of a plane and since the plane is infinite in extent, there are an infinite number of solutions. That is let
z = m
y = n
m and n from the integers and
x = 30 - 2n - 3 m
then
x + 2y + 3z = 30
 
Denis said:
AH AH: why did you use 0 then?:confused:
0 is not considered a positive integer (unless a change was recently made!)
Click to expand...
You did not get the word that 0 is now positive? You need to get out more often.
 
pka said:
@Jomo, Why did you think that I was addressing you? I was not.

Who did quote at the top of the posting? It had nothing to do with you.

I would have quoted your post if you were involved.
Click to expand...
pka, you did not address me at all. When I saw your solution I realized that you can have the variables take on negative values and I commented that I did not see that for some unknown reason.
 
Denis said:
Well, no matter how we cook it, there are 61 solutions
if x,y,z are integers > 0.
1: 1,1,9
2: 1,4,7
3: 1,7,5
....
59: 22,1,2
60: 23,2,1
61: 25,1,1
Click to expand...
You mean 61 positive integer solutions I presume. There are infinite integer solutions.
 
Denis said:
Yes; I did specify "if x,y,z are integers > 0".
Click to expand...
Sometimes I'm blind in one eye and can't see out of the other. Other times, well it's just another case of me sticking my big foot in my mouth. I'm not sure whether this time it might be a combination of the two. Please, pay it no mind. pretty please
 
Ishuda said:
Sometimes I'm blind in one eye and can't see out of the other. Other times, well it's just another case of me sticking my big foot in my mouth. I'm not sure whether this time it might be a combination of the two. Please, pay it no mind. pretty please
Click to expand...

Be careful - he might make you watch 5 hockey games, continuous one-after-the-other, as a penance........
 
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