How many integral values of x and y satisfy the eq. 7x-10y=13 so that 0<= x<= 1000?

Question

Please elaborate your method.

Answer 1

WOW!!!! I can see that 'elaborate your method' means something quite different to a mathematician than it does to the rest of you ☺

Since a Diophantine equation of the form ax+by=c has integer solutions if and only if the gcd(a,b) exactly divides c we check: gcd(7,-10) = 1 and 1 certainly divides 13 exactly (this step can save you a *lot* of time )

Now if 7x-10y=13 then
y=(7x-13)/10 and for y to be an integer, 7x-13 must be an exact multiple of 10. A bit of quick 'trial and error' shows that for x=9 then 7x-13=50 and y=5 so the first set of integers is

(9,5)

Now (from Euclids Algorithm) if x0 and y0 are any solutions of ax+by=c, then x=x0+bk and y=y0+ak are also solutions (where k = 0,±1, ±2, etc.). Since we're only interested in positive values for x (up to x≤1000) we'll only us positive k.

How many (x,y) values are there? Since x=x0+bk and b=10 and x0=9, there are int[(1000-9)/10] =99 more answers so a total of 100 answers.

Enumerating the first few

(9,5)
(19,12)
(29,19)
(39,26)
(49,33)
(59,40)
(69,47)
(79,54)
(89,61)
(99,68)
*
*
*
(999,698)


Doug

Answer 2

For 7x - 10y = 13,
integral values for (x, y), with 0 ≤ x ≤ 1000, are:
(1, 2), (11, 9), (21, 16), ... , (991, 695).
There are a total of 100 ordered pairs that satisfy the conditions in the problem.

Answer 3

7x-10y=13
10y = 7x-13

7x-13 mod 10 = 0 <=> 7x-3 mod 10 =0<=> -3x-3 mod 10 =0
<=> x=9 mod 10
x = 9 + 10k, y = (7(9+10k) - 13)/10; for k = 0,1,2,.....,99
k until 99 because for k=100 then x >1000
thus 100 possible x values

Answer 4

Answer is 100


7x -10y =13
x =10/7y+13/7
= y+3/7y+6/7+1 ..........................(1)

assuming y to be integer

3y/7 +6 /7 is integer
or

3y+6 is divisible by 7
y+2 is divisible by 7
y =7k-2
where k=1,2,3.....

using (1)

x =7k-2+1+3k
=10k-1

ie
x= 9,19,29 .....999

total 100 values for x
and for each value a y value is there

Answer 5

100integrals for x and the same number for y
general formula=(Xi,Yi)
Xi=10i-1
yi=7i-2
where i=1,2,........100
(x1,y1)=9,5
(x100,y100)=(999,698)