When are two given lines parallel and identical?

Question

So I need a bit help with lines. I am considering 2 cases,

1. When they are parallel and

2. When they are completely same, same coordinates, everything, one on top of another.

This is what I think how it is, please tell me if I'm right, and correct me if I'm wrong.

1. When are they parallel, for example if I have two lines \begin{align*} y_1&=m_1x+n_1,\\ y_2&=m_2x+n_2. \end{align*} and if $m_1 = m_2$ regardless of $n_1$ and $n_2$ they are parallel ?

2. When they are same (one on top of another) \begin{align*} y_1&=m_1x+n_1,\\ y_2&=m_2x+n_2. \end{align*} $m_2=m_1$ and $n_1=n_2$?

Image: a) Parallel b) Same lines

Answer 1

You're right. If $m_1=m_2$ the two lines are parallel, and if moreover $n_1=n_2$ the two lines are identical. On the left-hand side of your equation, you can just write $y$ instead of $y_1$ and $y_2$ (like you do with $x$).

Answer 2

You are right. Good work.

FYI: If you want to include vertical lines as well, then the lines will have the form \begin{align*} a_1 x + b_1 y &= c_1 \\ a_2 x + b_2 y &= c_2, \end{align*} where we assume $a_1^2 + b_1^2 = 1$ and $a_2^2 + b_2^2 = 1$. The two lines will be

• parallel if $a_1 b_2 = b_1 a_2$; and

• identical if $a_1 = a_2$, $b_1 = b_2$, $c_1 = c_2$.

Answer 3

they are parallel when their slopes m1 and m2 are equal, y-intercept n1 is not equal n2
y1=x+1 y1=x-1 slope =1 for both ; yint1=n1=1; yint2=n2=-1

same y1=x+1; y2=x+1 same equation