find positive integer solutions?

Question

find how many positive integer solutions does the equation 2x+3Y=100 have.

Answer 1

2x + 3y = 100
2x = 100 - 3y
x = (100 - 3y)/2
x = 50 - (3/2)y

- y must be divisible by 2 in order for x to be an integer.
and
- (3/2)y must be less than 50 for x to be positive
(3/2)y < 50
y < 100/3 < 34
and
- 0 is not a positive integer so you must subtract 1


So there are 34/2 - 1 solutions.
16 solutions.

Answer 2

Hmm. Interesting question.

Just looking at it, there are a few limits. X and Y must be greater than or equal to zero, and have no fractional part.

Next, look at 2X, and 3Y, if we divide 100 by 2, then the maximum X is 50. Likewise, the maximum Y is 33.

We can then think a little bit. 100-2x must be divisible by 3. That isn't so easy to work with. 100 - 3Y must be divisible by 2. Since 100 is even, that means that 3Y is even, so Y must be divisible by 2. Counting zero, there are 17 possible Y values. (0, 2, 4...32).

Now, 100-3Y always equals an even number, there is always some X value that can be used to make up that even number.

So you have it. There are 17 positive integer value pairs which make up the solution. If you omit zero values, there are 16 pairs of X and Y that solve the problem.

Answer 3

x y Answer
2 * 47 + 3 * 2 = 100
2 * 44+ 3* 4= 100
2* 41+ 3* 6= 100
2* 38+ 3* 8= 100
2* 35+ 3* 10= 100
2* 32+ 3* 12= 100
2* 29+ 3* 14= 100
2* 26+ 3* 16= 100
2* 23+ 3* 18= 100
2* 20+ 3* 20= 100
2* 17+ 3* 22= 100
2* 14+ 3* 24= 100
2* 11+ 3* 26= 100
2* 8+ 3* 28= 100
2* 5+ 3* 30= 100
2* 2+ 3* 32= 100
There are 16 combos of x and y that work.
However in one solution x equals y.

Answer 4

x = (100-3y)/2
As long as y is an even number, you have positive solutions.
Since 100-3y > 0, y can be 2, 4, ..., 32. Therefore, you have 16 solutions.
------
x = 50, y = 0 is not count.

Answer 5

Start with the ordered pair (2,32), you may get more ordered pairs by incrementing x by 3 and decrementing y by 2 since 2(3) - 3(2) = 0, it doesn't change the value of 100 at all.

So there are (5, 30), (8,28), (11, 26)... and so on until (47, 2) for a total of 16 pairs if I counted correctly.

Answer 6

x=50-3t and y =2t where t is a positive integer
so 3t<50 so t<17 so there are 16 positive solutions1<=t<=16
take as an example t =13
x= 11 and y = 26
22+78=100 OK

Answer 7

2x+3Y=100
3y = 100 - 2x
y = (100 - 2x) / 3

x ...... y
2 ..... 32
5 ..... 30
8 ..... 28
.....
.....
47 .... 2

X increase by 3
So: (47 - 2) / 3 +1 = 16 positive solutions
Or
Y decrease by (-2)
So: (2 - 32) / (-2) +1 = 16 positive solutions

Answer 8

17
From y =32 and x = 2 to y=2 and x = 47

Answer 9

there are 2 positive solutions right?