## 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

## On Equations of Intersecting, Coinciding, and Parallel Lines

When we have two lines on a plane, there are three possibilities:

• the lines will never meet (parallel)
• the lines will meet at one point (intersecting)
• the lines will meet at infinitely many points (coinciding).

As for the third case, coinciding means that lines which are on top of each other.

In Algebra, we have learned that a line can be represented with an equation. The equations which represent lines are called linear equations. We have learned that linear equations can be represented by $y = mx + b$, where $m$ and $b$ are real numbers.

We can examine the three cases mentioned above in terms of equations. Can we determine if lines are parallel, intersecting, or coinciding based on equations only?  » Read more