Finding integer soultion for a given equation is a heavily studied math topic. It is called diophantine equations you can look it up
simply 6y+2x=35 is classified as a linear diophanite equation
and it is trivial that this can not have a solution since 6y is even
and 2x is also even so there sum must be even since even+even=even .35 is odd so this equation can not have any integer soultion.actually the only way for these type of equation to have a soultion is that the gcd of the coefficent of x and the coefficent of y is divisible by the constant given after the = sign for example
97x+35y=13 the gcd(97,35)=1 since 1 is divisible by 13 then there exist soultions.note:here most high school level courses end here but if you are interested in finding the soultion . we now need to find some type of linear combination that of 97 and 35 that gives us 1 then multiply that equation by 13 so after some trial and error we find 97*13+35(-36)=1
if we multiply the equation by 13 ,97*169+35*(-468)=13 so our soultion are x=169 and y =-468 which are part of an infinite set of soultion by the
therom that states that if (x,y) are a soultion then it is part of a family of soultion in this form (x+ka,x-kb)where k is any integer and a is the coefficent of y and b is the coefficent of x so this equation has infinitely many soultions in the form(169+35k,-468-97k) where k is an integer
