# What are coincident lines?

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Mar 22, 2016

Well, I think you mean two lines that lie one on top of the other.

There is a slight difference between two parallel lines and two coincident lines.

Parallel lines have space between them while coincident don't. Parallel lines do not have points in common while coincident ones have ALL points in common!!!

When you consider the mathematical form $y = a x + b$ for your lines you have:

1) Parallel lines differs only in the real number $b$ and have the same $a$ (slope).

For example:
The two lines:
$y = 3 x + 3$ and $y = 3 x + 5$ are parallel.
The two lines described by these equations have the same inclination but cross the $y$ axis in different points;

2) Coincident lines have the same $a$ and $b$.
Sometimes can be difficult to spot them if the equation is in implicit form: $a x + b y = c$.

For example:
The two lines:
$x + y = 3$ and $2 x + 2 y = 6$ are coincident!!! (Basically the second is the first multiplied by $2$!!!). This situation happens frequently in Linear Algebra when you solve systems of linear equations.

If you isolate $y$ on one side you'll find that are the same!!!
Try to plot them and see.

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