## A Closer Look at Coinciding Lines

In the previous post, we have asked a question about coinciding lines. We observed that the lines with equations $3x + 8y = 12$ and $6x + 16y = 24$ coincide. It is not difficult to see that $6x + 16y = 24$ is $2(3x + 8y = 12)$. The question now is if one equation is a multiple of the other, are their graphs coinciding? We answer this question below.

Consider a point with coordinates (2,3). What happens if we multiply the coordinates by 2, 3, and 4? If we do this, the coordinates become (4, 6), (6,9), and (8,12). Now, what is so special about these points? As we can see in the graph below, they lie on the same line. Can you explain why?  » Read more