All lines with equation ax+by=c such that a, b, c form an arithmetic progression pass through a common point. What are the coordinates of that point?

Answer 1

The answer is (-1,2), the question is how to determine that (or even prove it). A pretty cool problem.

Try determining it by making up specific examples.

You might try x+2y = 3 (1,2,3 is an arithmetic progression with a difference of 1).

The 2nd equation might be 5x+7y = 9. Now solve this system of equations using elimination or substitution method. Try this with a few different specific examples, just to convince yourself.

Now how do you actually prove it?

Try this.

!st equation: ax + (a+d)y = a+2d (common difference is d)

2nd equation: bx + (b+k)y = b+2k (common difference is k)

Use elimination method by multiplying first equation by -b and 2nd equation by a, then add to eliminate x. You will find that this simplifies to y = 2. Plug 2 back into one of the equations for y, and solve for x. Result: x= -1.

Also you might try using Desmos graphing calculator graphing several lines with specified criteria, along with the point (-1,2).