In a linear programming problem, I encountered a situation where the maximum value of independent variables are fraction at an angular point [Variables are such that the can't be fraction].Then my teacher told me to find the values of variables in the straight lines of the graphs...
My question is, how do we know that which points satisfy the given conditions?There are innumerable in the lines...
Integer programming, i.e., linear programming with the added constraint that the solutions should be integer points, is a hard problem compared to linear programming, which is completely solved.
In practice, you're going to want to pick integer points close to your discovered max point, by rounding the rational values to nearest integers, maybe $\pm 1$ to cover more ground, eliminating integer points that don't satisfy the constraints, and testing the remaining ones in the object function.
For example, with a 2-variable problem, if you got the solution $(19/2,40/3)$, you could look at integer points with $x=9$ or $10$, and $y=13$ or $14$. That's four ordered pairs: $(9,13),(9,14),(10,13),(10,14)$. It's possible that some of those aren't feasible solutions, so you can eliminate those. Whichever ones are feasible, you can test. To search a little further, you might try points with $x$-values $8$ or $11$, and similarly for $y$.
This method is not guaranteed to discover the optimal answer, but it will work in many cases.
If you can draw a picture of the feasible region, including a few lines indicating fixed values of the object function, then you can probably work it out visually.