# Parallel Line Calculator

URL: https://speedycalculator.com/calculators/math/parallel-line/
Last updated: 2026-10-30

> The parallel line calculator finds the line parallel to y = mx + r that passes through a given point, and the distance between the two parallel lines.

## Inputs

- First line: slope m
- First line: intercept r
- Point – x₀
- Point – y₀

## Results

- Parallel line slope a
- Parallel line intercept b
- Distance between the lines

## Formula

```
Parallel line: y = m·x + b with b = y₀ − m·x₀
Distance = |b − r| ÷ √(m² + 1)
```

## Explanation

Parallel lines have the same slope and never meet. Drawing a parallel through a point is a basic construction in geometry, engineering drawings and road design.

### How to use the Parallel Line Calculator

- **First line: slope m**
- **First line: intercept r**
- **Point – x₀**
- **Point – y₀**

Results update as you type, and everything is calculated in your browser.

### How it works

The slope is copied from the given line; the intercept is chosen so that the line passes through the point. The perpendicular distance between the lines follows from the intercepts.

### Inputs explained

FieldWhat to enterFirst line: slope mA numberFirst line: intercept rA numberPoint – x₀A numberPoint – y₀A number

### What each result means

- **Parallel line intercept b** = [Point – y₀] - [First line: slope m]×[Point – x₀]
- **Distance between the lines** = abs([Point – y₀] - [First line: slope m]×[Point – x₀] - [First line: intercept r])÷sqrt([First line: slope m]^2 + 1)

### Example values

How the main result changes when **First line: slope m** changes and everything else stays at the example values above:

First line: slope mParallel line slope a111.51.5222.52.53344

## Worked example

Inputs — First line: slope m: 2; First line: intercept r: 1; Point – x₀: 3; Point – y₀: -2.

Results — **Parallel line slope a = 2**; Parallel line intercept b = -8; Distance between the lines = 4.02492236.

## Key terms

- **Parallel lines**: Lines in a plane that never meet.

## FAQ

**How do I know two lines are parallel?**
They have the same slope and different intercepts.

**What is the parallel line through (3, −2) to y = 2x + 1?**
y = 2x − 8.

**What about vertical lines?**
They are parallel when they have the form x = constant.

**What if the point is on the original line?**
The "parallel" line is the same line and the distance is 0.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)