# Equations of Parallel Lines

## Key Questions

• Choose any two distinct points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ on the line then use the slope formula below to find the slope $m$.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$.

I hope that this was helpful.

• Parallel lines are lines that never intersect. Because of this, a pair of parallel lines have to have the same slope, but different intercepts (if they had the same intercepts, they would be identical lines).

So, to find an equation of a line that is parallel to another, you have to make sure both equations have the same slope. In the general equation of a line $y = m x + b$ , the $m$ represents your slope value.

An example of paralell lines would therefore be:

(1) $y = m x + b$
(2) $y = m x + c$

With $b$ and $c$ being any constants. Note that they have to be different, because if they were equal, then you'd just have two identical lines that technically intersect in every single point.

Sometimes though, linear equations aren't in the form $y = m x + b$. You could have something like:

$8 x + 2 y = 16$

Here, you can't directly pick out the slope. But you could always turn that into the form $y = m x + b$ to find your slope $m$ by simply solving for $y$.

$8 x + 2 y = 16$
$2 y = 16 - 8 x$
$y = 8 - 4 x = - 4 x + 8$
There we go. We can see that $m = - 4$ here. So to write a line parallel to that, we have the general equation of

$y = - 4 x +$ any constant except for $8$.

• Since if slopes of lines are different, then they will intersect at some point, parallel lines must have the same slope.

I hope that this was helpful.

• If distinct lines have the same slopes, then they are parallel.

I hope that this was helpful.