When using the method of

elimination, after how many steps beyond multiplying both

equations by two multipliers to put the equations in forms ready for elimination

method, you would know for sure that the lines are going to be parallel and

there is no solution to the system of equations?

2. Answer the

same question for infinitely many solutions.

I am not exactly sure what you are asking, but I will try to answer it from what I understand.

If you are solving a system of equations by elimination method, you will be able to determine if the system has one solution, no solutions, or infinitely many solutions by looking at the coefficients on x and y before adding them. If both x and y are going to cancel out, then you have either no solution or infinitely many solutions. If the constant on the right are going to cancel out (same number with opposite signs) then there are infinitely many solutions (same line). If the constants are not going to cancel out, then there is no solution (parallel lines).