parallel lines

How to find the slope of a parallel line?

Every straight line in a two-dimensional space can be described by a simple line equation:

𝑦=𝑎𝑥+𝑏,y=ax+b,

where a and b are coefficients, x is the x-coordinate, and y is the y-coordinate. Every line is uniquely defined if the values of a and b are known.

Let’s assume that you know the following information:

1. The equation of the given line is y = mx + r. You know the values of m and r and are looking for a line parallel to this one.
2. You also know the coordinates of the point your line is supposed to pass through. They are x₀ and y₀.

The slope of any line is equal to the value of a coefficient. If two lines are parallel, then they must have the same slope. From this, we can deduce that.

a = m

Parallel line equation

Once you know the a coefficient of the line, all that is left to do is determine the b coefficient (also known as the y-intercept).

The method is straightforward: you have to substitute the coordinates (x₀, y₀) and the value of a into the equation of your line.

𝑦=𝑎𝑥+𝑏y=ax+b

𝑦₀=𝑚𝑥₀+𝑏y₀=mx₀+b

𝑏=𝑦₀−𝑚𝑥₀b=y₀−mx₀

Finding the distance between two parallel lines

Now that you know the equation of your new line, you can easily use it to determine the distance between it and the first line. In this case, the distance is defined as the length of the shortest possible segment that would join the two lines together.

Our parallel line calculator finds this distance automatically. If, however, you would like to check whether the result is correct, you can use the distance formula:

$D=\mid b-r\mid m^2+1D=m^2+1\mid b-r\mid$

Parallel line calculations: an example

If you’re still not sure how to find the equation of a parallel line, take a look at the example below!

1. Write down the equation of the first line. Let’s say it’s:

   y = 3x - 5

2. Write down the coordinates of the given point P that the second line will pass through. Let’s assume it is (1,6). In other words, x₀ = 1 and y₀ = 6.

3. Write down the equation of your new line: y = ax + b. You will try to determine the values of coefficients a and b.

4. Coefficient a is equal to m. Hence,

   a = m = 3.

5. Plug the coordinates of point P into the equation of your new line to determine b:

   y₀ = ax₀ + b

   6 = 3 × 1 + b

   b = 6 - 3 × 1 = 3

6. Knowing the values of the slope and y-intercept, you can now write down the full equation of the new line: y = 3x + 3.

7. You can also calculate the distance between the two lines:

   $D=|b-r|/\sqrt{m^2+1}$

   $D=|3-6|/\sqrt{3^2+1}=|-3|/\sqrt{10}=2.53$

The distance between the two lines is equal to 2.53.

FAQ

How do I calculate the distance between two parallel lines?

To find the distance between two parallel lines in the Cartesian plane, follow these easy steps:

1. Find the equation of the first line: y = m1 × x + c1.
2. Find the equation of the second line y = m2 × x + c2.
3. Calculate the difference between the intercepts: (c2 − c1).
4. Divide this result by the following quantity: sqrt(m² − 1):
   d = (c2 − c1) / √(m² − 1)

This is the distance between the two parallel lines.

How do I identify two parallel lines on the Cartesian plane?

Two parallel lines on the Cartesian plane have the same angular coefficient. Knowing this, you can exclude the trivial case where the intercept is the same (the two lines coincide).

It’s harder to define parallelism in three dimensions: two lines can be nonparallel yet never intersect!

How do I find the parallel line passing through a point?

The parallel to the line y = m × x + b passing through the point (p, q) can be found with the following steps:

1. Find the slope of the parallel line — it’s m!
2. Find the intercept of the parallel line’s equation substituting the coordinates of the point in the original line’s equation:
   q = m × p + b
   Hence:
   b = q / (m × p)

What are some examples of parallel lines?

The sides of a road are an excellent example of parallel lines: before an intersection, the two sides never meet! Other examples are the parallels (but not the meridians!) on a globe: those are three-dimensional parallel lines. You can find parallel lines on many human creations, but rarely in nature: with a keen eye, you can see them in some geological formations and for brief lengths in trees and other plants.