Find all solutions with x and y positive 5x 6y 100 6x 7y

Find all solutions with x and y positive. 5x + 6y = 100 6x + 7y = 200

Solution

13.

gcd(5,6)=1

6-5=1

MUltiplyig by 100 gives

6*100-5*100=100

IN general

6*(100+5n)+5*(-100-6n)=100

We want:100+5n>0

5n>-100

n>-20

-100-6n>0

100+6n<0

6n<-100

HEnce,

n<=-17

So, n can take values

-17,-18,-19

Hence, 3 solution

(x,y)=(100+5n,-100-6n)

n=-17

(x,y)=(15,2)

And so on for each value of n

14.

gcd(7,6)=1

7-6=1

MUltiplyig by 200 gives

7*200-6*200=200

IN general

7*(200+6n)+6*(-200-7n)=200

We want:200+6n>0

3n>-100

n>-33

n>=-32

-200-7n>0

200+7n<0

7n<-200

n<-28.57

HEnce,

n<=-29

So, n can take values

-29,-30,-31,-32

Hence, 4 solution

(x,y)=(200+6n,-200-7n) for each n