Simultaneous Equations

I was thinking about how to produce simultaneous equation questions with 'nice' numbers. Then I realised that I only had one sure way of producing a question. For example:

• I pick an equation x + y = 4
• I know that perpendicular lines to this are such that the product of their gradients is -1. So required line is y - x = c
• Choose a point on my chosen line, (- 2,6).
• Required line has c = 6 - (- 2) = 8
• Giving the other line as y - x = 8.

But there are lots, possibly infinite, lines that pass through this point, i.e. (- 2, 6). How are they defined?

This is something I've never considered before. (Is this 'across the grain' thinking?) I want these equations to have whole number co-efficients. Now I need to investigate and see if I can find a way of classifying them.