Equation of a Parallel Line Calculator

Calculator

Choose a method, enter the given line, then set the point for your parallel line.

Inputs accept decimals or fractions like 3/4.

Method

Standard (A, B, C) Slope / Vertical Two Points

Given line in standard form

A

B

C

This represents: Ax + By + C = 0. Parallel lines keep the same A and B.

Given line by slope, or a vertical line

Line type

Slope (m)

If you only know a slope, the intercept is solved using the point below.

Two points on the given line

x₁

y₁

x₂

y₂

If x₁ = x₂, the given line is vertical and the parallel line will be x = xₚ.

Point for the parallel line

xₚ

yₚ

The new line must pass through this point and remain parallel.

Reset

Example data table

Use these examples to test the calculator quickly.

Given line	Method	Point (xₚ, yₚ)	Parallel line (result)
2x − 3y + 6 = 0	Standard	(3, −2)	2x − 3y − 12 = 0
y = −1/2 x + 4	Slope	(8, 1)	y = −1/2 x + 5
Through (1, 2) and (5, 4)	Two Points	(0, −1)	y = 1/2 x − 1
x = 4	Vertical	(9, 10)	x = 9

Formula used

Parallel lines share the same slope (or the same normal direction).

1) From standard form

Ax + By + C = 0

A parallel line keeps A and B:

Ax + By + C′ = 0

Using point (xₚ, yₚ):

C′ = −(A xₚ + B yₚ)

2) From slope

Non-vertical line:

y = m x + b

Parallel line has the same m:

b′ = yₚ − m xₚ

y = m x + b′

Vertical line:

x = k ⇒ x = xₚ

How to use this calculator

Select how your given line is described: standard, slope, or two points.
Enter the required values for that method.
Enter the point (xₚ, yₚ) where the parallel line must pass.
Press Calculate to view results above the form.
Use the download buttons to export your result.

What this parallel line calculator does

This calculator builds the equation of a line that is parallel to a given line and passes through a chosen point (x_p, y_p). You can enter the given line in standard form, by slope, or by two points. The output is shown in multiple equation styles so you can copy the form you need for classwork, graphs, or reports.

Parallel lines and equal slope

Two non-vertical lines are parallel when their slopes are equal. If a line has slope m, every parallel line also has slope m and differs only by its intercept. For vertical lines, slope is undefined, and parallelism means both lines have the form x = constant. This tool detects vertical cases automatically.

Using standard form Ax + By + C = 0

In standard form, the vector (A, B) is a normal direction. Parallel lines keep the same A and B because the normal direction does not change. Only the constant term changes. After you supply (x_p, y_p), the new constant is computed as C′ = −(A x_p + B y_p) so the point satisfies Ax + By + C′ = 0.

Using slope input and vertical lines

If you enter a slope m for the given line, the calculator keeps that m and solves the intercept with b′ = y_p − m x_p. If the given line is vertical (x = k), the parallel line must also be vertical, and the only vertical line through the point is x = x_p.

Using two-point input

Two points define a direction. The calculator computes Δx and Δy from (x₁, y₁) and (x₂, y₂). If Δx = 0, the given line is vertical. Otherwise the slope is m = Δy/Δx, then the parallel line uses the same m and passes through (x_p, y_p).

Interpreting the output forms

You get slope-intercept form (y = mx + b), point-slope form (y − y_p = m(x − x_p)), and standard form. Standard form is also normalized to a consistent scale for comparison. Use point-slope when you want the “through a point” structure, and slope-intercept when graphing quickly.

Common checks and practical uses

A quick check is to substitute (x_p, y_p) into the final equation and confirm it balances. For non-vertical lines, verify the slope matches the given line’s slope. Parallel-line equations appear in geometry proofs, coordinate graphing, engineering sketches, and aligning offsets in design layouts. Exporting to CSV or PDF helps keep records consistent.

FAQs

1) What makes two lines parallel?

Non-vertical lines are parallel when they have the same slope. Vertical lines are parallel when both have the form x = constant. Parallel lines never intersect.

2) Why does standard form keep A and B the same?

In Ax + By + C = 0, the pair (A, B) defines the line’s normal direction. Parallel lines have the same direction, so A and B remain unchanged while only C shifts.

3) How do I find the new intercept b?

For a non-vertical line with slope m through (x_p, y_p), use b = y_p − m x_p. The calculator applies this automatically.

4) What happens if the given line is vertical?

If the given line is x = k, the parallel line must also be vertical. The only vertical line passing through your point is x = x_p, regardless of y_p.

5) Why show point-slope and slope-intercept together?

Point-slope emphasizes “through a point” and is great for derivations. Slope-intercept is easiest for graphing and quick reading. Both represent the same line.

6) Can I enter fractions like 3/4?

Yes. You can enter decimals or fractions in many fields. The calculator evaluates them and shows clean numeric output where possible, including helpful approximations for repeating decimals.