Find the number positive integer solutions of
4x+6y=200

- mathmath333

Find the number positive integer solutions of
4x+6y=200

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- mathmath333

\(\large \color{black}{\begin{align}
& \normalsize \text{Find the number positive integer solutions of}\hspace{.33em}\\~\\
& 4x+6y=200
\end{align}}\)

- mathmath333

I need to find quick way

- Empty

I don't really know but you should look into the Euclidean Algorithm I think that might be a good path to start looking into.

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## More answers

- anonymous

as this is Diophantine equation i agree with @Empty

- anonymous

so lets try
2x+3y=100

- ganeshie8

yeah both equations are same but we don't really need euclid algorithm here as finding a particular solution seems kinda easy by inspection.
By inspection, (50, 0) is one particular solution

- anonymous

but Euclid would help in finding general solution to show it's infinite solutions right ?

- ganeshie8

euclid is for finding particular solution
since we already figured that (50, 0) is a particular soltion, we can avoid euclid

- anonymous

:3

- mathmath333

I found out (50,0),(47,2),.....(2,32) but that took more than 2 min

- ganeshie8

finding "one" particular solution is enough

- anonymous

anyway i like this way to show its infinite:-
|dw:1442748565442:dw|

- ganeshie8

after you have one particular solution, try finding the "null" solution :
2x + 3y = 0
Easy to see that (-3, 2) solves above equation.
Therefore the complete solution is given by : `(50, 0) + t(-3, 2)`

- ganeshie8

since you want just the positive integer solutions, solve :
50 - 3t > 0
0 + 2t > 0

- mathmath333

50/3=16.

- mathmath333

answer is 17

- ganeshie8

right, solve it simultaneously
you should get an interval of "t" as solution

- anonymous

aha i haven't note positive :O

- ganeshie8

I think the answer should be 16

- mathmath333

0

- ganeshie8

nope
0 < t < 16.66
there are exactly 16 positive integers in that interval

- mathmath333

0

- ganeshie8

0 < t < 50/3
leave it like that

- mathmath333

ok u r right (50,0) doesn't count answer is 16

- ganeshie8

Yep! lets do one more example maybe ?

- ganeshie8

Find the number of positive integer solutions to the equation
7x + 13y = 700

- ganeshie8

If you prefer, here are the steps :
1) Find any one `particular` solution by inspection
2) Find the `null` solution
3) Write out the complete solution : `particular` + `null`

- mathmath333

(100,0)

- ganeshie8

Yep, keep going

- mathmath333

how to find null soln

- ganeshie8

as the name says, it is the solution to the equation
7x + 13y = 0

- ganeshie8

give it a try.. it would feel awesome if you figure out a method to find the null solution on ur own..

- mathmath333

(-13,-7)

- ganeshie8

Very close, but no.
plug them in and see if they really produce 0

- mathmath333

(-13,7)

- ganeshie8

Excellent! that is one null solution.
Notice that any multiple of that also works,
so all the null solutions are given by ` t(-13, 7)`
where ` t ` belongs to the set of integers

- mathmath333

100-13t>0
0+7t>0

- ganeshie8

Yes, you have skipped step3 but ok..

- ganeshie8

go ahead find the total count

- mathmath333

what is step 3 ?

- mathmath333

oh this one
"3) Write out the complete solution : particular + null"

- ganeshie8

Yes, I was refering to that..

- mathmath333

7 is answer

- ganeshie8

Yep! congratulations!
Now you know how to solve any linear diophantine equation of form \(ax+by=c\)

- mathmath333

cool!

- ganeshie8

so what was your method for finding null solution ?

- mathmath333

ax + by = 0
(-b,a)

- ganeshie8

thats it!

- ganeshie8

that works always!
so finding null solution is trivial
as you can seethe only hard part is finding a particular solution

- anonymous

6y=200-4x
3y=100-2x
\[y=\frac{ 100-2x }{ 3 }\]
by hit and trial
when x=2
\[y=\frac{ 100-4 }{ 3 }=32\]
add successively 3 to the value of x
x=2+3=5,y=30
x=5+3=8,y=28
x=11,y=26
....
x=47,y=2
x=50,y=0

- anonymous

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