Does the equatoin 2x+6y=35 have any whole number solutions? Why or why not?

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