Slopes of parallel and perpendicular lines calculator

Created by Bogna Szyk

Reviewed by Adena Benn

Last updated: Jul 11, 2022

If you're scratching your head while trying to figure out some parallel lines equations, stop worrying: this parallel line calculator is precisely the tool you need. In just a few seconds, it will determine the equation of a line that is parallel to a given line and passes through a given point. That's not all, though: our calculator is also able to find out what's the distance between the two lines.

Read on to discover how to find the slope of a parallel line, or what a y-intercept is. Also, make sure to check out the perpendicular line calculator!

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

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

We also recommend checking our average rate of change calculator.

Parallel line equation

Once that 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 + b

y₀ = m * x₀ + b

b = y₀ - m * x₀

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 = |b - r| / √(m² + 1)

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.

1. 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

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

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

D = |b - r| / √(m² + 1)

D = |3 - 6| / √(3² + 1) = |-3| / √(10) = 2.53

The distance between the two lines is equal to 2.53.

First line equation y = mx + r

Second line passes through point...

Parallel line equation y = ax + b

Distance between the lines

Average rate of changeBilinear interpolationCatenary curve… 35 more

Once you know the equation of the new line, finding the intersection point between it and the first (given) line is a straightforward task. All you have to do is find a point with coordinates (xₐ,yₐ) such that it lies on each of the two lines.

Consider the example we've just analyzed. We found two perpendicular lines: y = 2x - 2 and y = -0.5x + 6.5. These two equations form a system of equations with two unknowns - the coordinates of the point of intersection.

Let's solve this system of equations:

yₐ = 2xₐ - 2

yₐ = -0.5xₐ + 6.5

Multiplying the second equation by 4, you get

yₐ = 2xₐ - 2

4yₐ = -2xₐ + 26

Adding the two equations together,

5yₐ = 24

From there,

yₐ = 4.8

xₐ = 0.5yₐ + 1 = 2.4 + 1 = 3.4

The coordinates of the point of intersection are (3.4, 4.8).

Of course, you don't have to carry out these tedious calculations all by yourself - our perpendicular line calculator can do the same in just a few seconds! And don't forget to check our other coordinate geometry calculators like the average rate of change (aroc) calculator.

What is the slope of a perpendicular line?

How do you find the slope of a perpendicular slope?

How do you know if 2 slopes are perpendicular?