mathmath333
  • mathmath333
Find the number positive integer solutions of 4x+6y=200
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{Find the number positive integer solutions of}\hspace{.33em}\\~\\ & 4x+6y=200 \end{align}}\)
mathmath333
  • mathmath333
I need to find quick way
Empty
  • 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.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

anonymous
  • anonymous
as this is Diophantine equation i agree with @Empty
anonymous
  • anonymous
so lets try 2x+3y=100
ganeshie8
  • 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
  • anonymous
but Euclid would help in finding general solution to show it's infinite solutions right ?
ganeshie8
  • ganeshie8
euclid is for finding particular solution since we already figured that (50, 0) is a particular soltion, we can avoid euclid
anonymous
  • anonymous
:3
mathmath333
  • mathmath333
I found out (50,0),(47,2),.....(2,32) but that took more than 2 min
ganeshie8
  • ganeshie8
finding "one" particular solution is enough
anonymous
  • anonymous
anyway i like this way to show its infinite:- |dw:1442748565442:dw|
ganeshie8
  • 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
  • ganeshie8
since you want just the positive integer solutions, solve : 50 - 3t > 0 0 + 2t > 0
mathmath333
  • mathmath333
50/3=16.
mathmath333
  • mathmath333
answer is 17
ganeshie8
  • ganeshie8
right, solve it simultaneously you should get an interval of "t" as solution
anonymous
  • anonymous
aha i haven't note positive :O
ganeshie8
  • ganeshie8
I think the answer should be 16
mathmath333
  • mathmath333
0
ganeshie8
  • ganeshie8
nope 0 < t < 16.66 there are exactly 16 positive integers in that interval
mathmath333
  • mathmath333
0
ganeshie8
  • ganeshie8
0 < t < 50/3 leave it like that
mathmath333
  • mathmath333
ok u r right (50,0) doesn't count answer is 16
ganeshie8
  • ganeshie8
Yep! lets do one more example maybe ?
ganeshie8
  • ganeshie8
Find the number of positive integer solutions to the equation 7x + 13y = 700
ganeshie8
  • 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
  • mathmath333
(100,0)
ganeshie8
  • ganeshie8
Yep, keep going
mathmath333
  • mathmath333
how to find null soln
ganeshie8
  • ganeshie8
as the name says, it is the solution to the equation 7x + 13y = 0
ganeshie8
  • ganeshie8
give it a try.. it would feel awesome if you figure out a method to find the null solution on ur own..
mathmath333
  • mathmath333
(-13,-7)
ganeshie8
  • ganeshie8
Very close, but no. plug them in and see if they really produce 0
mathmath333
  • mathmath333
(-13,7)
ganeshie8
  • 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
  • mathmath333
100-13t>0 0+7t>0
ganeshie8
  • ganeshie8
Yes, you have skipped step3 but ok..
ganeshie8
  • ganeshie8
go ahead find the total count
mathmath333
  • mathmath333
what is step 3 ?
mathmath333
  • mathmath333
oh this one "3) Write out the complete solution : particular + null"
ganeshie8
  • ganeshie8
Yes, I was refering to that..
mathmath333
  • mathmath333
7 is answer
ganeshie8
  • ganeshie8
Yep! congratulations! Now you know how to solve any linear diophantine equation of form \(ax+by=c\)
mathmath333
  • mathmath333
cool!
ganeshie8
  • ganeshie8
so what was your method for finding null solution ?
mathmath333
  • mathmath333
ax + by = 0 (-b,a)
ganeshie8
  • ganeshie8
thats it!
ganeshie8
  • ganeshie8
that works always! so finding null solution is trivial as you can seethe only hard part is finding a particular solution
anonymous
  • 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
  • anonymous
.

Looking for something else?

Not the answer you are looking for? Search for more explanations.