Question 137402
I am not sure what you mean by corresponding. If you mean, how can you tell if two lines are really the same line, then one equation will be a simple multiple of the other one.

5x + 3y = 10    two equations are multiples, so they produce the same graph
15x + 9y = 30

If you asking how to tell if two lines are parallel without finding the slope, then you would need to look at the ratio of both equations x and y coefficients. I suppose that is less work than converting to slope-intercept. But unless speed is the requirement, it seems like more effort than it is probably worth.

5x + 3y = 10    ratio is 3/5
15x + 9y = 10   ratio is 9/15 = 3/5. different graph, but parallel

hope that helps some