I've been having trouble getting started with this problem.
Suppose $x_1,x_2,x_3$ are integers $\geq 0$, satisfying $$21.7x_1-18.2x_2-19.4x_3=5.3$$
Then show $$7x_1+8x_2+6x_3=3+10z_1$$.
(*edit*)where $x_1,x_2,x_3,z_1$ are all integers $\geq 0$
This is from Joel Franklin's Methods of Mathematical Economics page 131.
my attempt: I multiplied the constraint by 10 to get $217x_1-182x_2-194x_3=53$. I found my first basic optimal solution with $x_1=53/217$. This left me with Gomory cut with
$\frac{35}{217} x_2+ \frac{23}{217} x_3 - z_1= \frac{53}{217}$ (where the inverse of inverse of basis matrix was 1/217, and I had to take the positive complements of -182/217 and -194/217 for $x_2,x_3$) as an added constraint.
I am not sure what to do next. Any help would be appreciated. Thanks.
Multiply the first equation by $10$ as you already did and from this subtract the second equation you need to show. Simplify and rearrange to express $z_1$ in terms of $x_1,x_2,x_3$ and constants. It is easy to show that $z_1$ is a sum of products of integers, and hence itself an integer. The part $z_1\ge 0$ follows from the second equation.