Find the equation of a parallel line calculator

Instructions: Use this tool to determine, showing all the steps, whether or not the two provided lines are parallel. Please type two linear equations in the boxes provided.

More about this parallel lines calculator tool.

Geometrically speaking, two lines are parallel when they don't intersect, or they potentially are the same line. So then, if you graph the two lines, you will visually see that they don't intersect. But that can be tricky.

But naturally, there are algebraic ways to assess whether or not two lines are parallel. One of the simplest way is to use the slope criterion.

How to determine if two lines are Parallel?

There are a couple of ways:

1. Graphically: Take a look at the graph, and if the lines don't intersect, then the lines are parallel
2. Algebraically: Compute the slope to each of the lines. If they have the same slope, the lines are parallel

The advantage of the graphical method is that it is simple, and it involves just taking a look at the graph, but naturally, in order to do so, you need to construct the graphs.

A disadvantage of the graphical method is that your eyes can deceive you. It may seems that the graph of lines do not intersect, but perhaps you are not graphing a big enough portion of the line.

An advantage of the algebraic method is that it is unequivocal. If the slopes coincide, the lines are parallel, and if they don't, then the lines are not parallel.

The one disadvantage of the algebraic method is that you need to take the work of formally computing the slope.

Parallel line equation

Observe that lines that are parallel will have the same slope. So if the equation of a line is \(y = a x + b\), what is the equation of the parallel line?

First, there is not one parallel line, there are actually infinite parallel lines, and the equation is \(y = a x + c\), for any \(c\).

As we can see, \(y = a x + b\) and \(y = a x + c\) both have a slope equal to "a", so they are parallel.

If you don't have the lines already in slope-intercept format, you can always solve for y, or solve for x is you want to reverse the axes.

The slope criterion

Two lines are parallel if they have the same slope. So that is the easiest way to go about determining if two lines are parallel, you simply compute the slope of both lines and check whether they are the same or not.

One exception is the case of two vertical lines, which are parallel, although we cannot compare the slopes because they are undefined.

If you have something like the slope-intercept form of the lines already given, you can directly assess whether or not the lines are parallel. Otherwise, you need the extra step of computing the slopes before comparing them.

Geometric interpretation of the graph of two parallel lines

Two lines that are parallel correspond to a system of equations with no solutions (or infinite solutions), where each equation represent one line.

Also, when the lines are not parallel they will intersect at one point and at one point only, which corresponds to a system of equations with a unique solution.

Example

Determine whether or the lines \(2x + 3y = 1\) and \(x + y = 3\) are parallel.

Answer:

First Line: Put the first equation in slope-intercept form

We have been provided with the following equation:

\[\displaystyle 2x+3y=1\]

Putting \(y\) on the left hand side and \(x\) and the constant on the right hand side we get

\[\displaystyle 3y = -2x +1\]

Now, solving for \(y\), the following is obtained

\[\displaystyle y=\frac{-2}{3}x+\frac{1}{3}\]

and simplifying all the terms that need simplification, we finally get the following

\[\displaystyle y=-\frac{2}{3}x+\frac{1}{3}\]

Second Line: Put the second equation in slope-intercept form

We have been provided with the following equation:

\[\displaystyle x+y=3\]

Putting \(y\) on the left hand side and \(x\) and the constant on the right hand side we get

\[\displaystyle y = -x +3\]

Analyze and Compare Slopes

Based on this information, we find that the slope of the first line is \(m_1 = -\frac{2}{3}\) and the slope of the second line is also \(m_2 = -1\), which are unequal, so then the lines are NOT parallel.

Notice that if perpendicularity is what you are looking for, you can use this perpendicular line calculator. By definition, perpendicular lines are not parallel, because perpendicular lines have different slopes ALWAYS.

An online parallel and perpendicular line calculator will help you to determine the equation of the perpendicular and parallel to the given line that passes through the entered points. Also, this free parallel line calculator provides a step-by-step solution for particular equations. Let’s take a look to get information about parallel and perpendicular lines.

What is a Perpendicular Line and a Parallel Line?

In mathematics, the perpendicular lines intersect each other at right angles. In simple words, a line is a perpendicular, when one line meets another at 90 degrees.

Properties of Perpendicular Lines:

• Perpendicular lines always intersect at 90°.
• All perpendicular lines can be considered as intersecting lines, but all intersecting lines cannot be termed as perpendicular because they must intersect at right angles.

Also, you can find the equation of the line that is perpendicular to this line and passes through the point by using the free online perpendicular lines calculator.

Conversely, when two lines will never intersect to each other is called parallel lines. The lines maintain to the same distance and never touch all the time.

Properties of Parallel Lines:

• Parallel lines are still equidistant from each other.
• These lines never meet at any common point.
• They always lie in the same plane.

Our free find the equation of the line that is parallel to this line and passes through the point calculator is simple and fast to perform calculations.

For instance, the adjacent sides of the square remain perpendicular to each other. However, if two lines are perpendicular to the same line, then these two lines will are parallel and never intersect to each other. However, you can perform the calculations related to the equation of the line segment with the assistance of this online perpendicular and parallel line calculator.

Perpendicular Line Equation with Examples:

Example #1:

Suppose a line passes through the point (5, 8) and perpendicular to the line \(y = 3x – 6\). We can find the perpendicular equation by following these steps:

Solution: 

Identify the slope (j) and the y-intercept (k) of given line. So, j = 3 and k = -6.

Compute the slope of the line which, is equal to\( a = -1/j = -1/3 = -0.334\)

Substitute the value of a in line equation\( y = ax + b : y = -0.334x + b\)

Plug the coordinates (5, 8) for x and y:

$$8 = -0.334 * 5 + b$$

$$8 = – 1.67 + b$$

$$b = 9.67$$

So, put the value of b in line equation:

$$y = – 0.334x + 9.67$$

When you put the same values in the perpendicular line calculator it’ll provide the same values.

Example #2: 

Consider one line passes through the points (0, –3) and (–1, –7) and another line passes through the points (3, 0) and (–5, 2). Are these lines parallel or perpendicular?
Solution:
The slope of first line,
m1 = (-7+3)/-1 = -4/-1 = 4
m2 = (2-0)/(-3-5) = 2/(-8) = -¼
Since, m1 ≠ m2, hence, lines are not parallel.
m1.m2 = 3 x (-⅓) = -1
Therefore, the two lines are perpendicular.
You can check the above results with the help of determine if the lines are parallel or perpendicular calculator

However, an Online Slope Calculator helps to find the slope (m) or gradient between two points in the Cartesian coordinate plane. Besides that, if you calculate the coordinate of a plan in a given line, you can use a coordinate that makes segment parallel/perpendicular calculator.

How to Find the Slope of a Perpendicular Line?

By a simple line equation:

$$y = ax + b$$

Where x is the x-coordinate, “a” and “b” are coefficients, and y is the y-coordinate.

Suppose that you already know the following information:

The equation of the line is \(y = mx + r\) which is used by perpendicular line calculator. And the coordinates of the point of the line are \(x_j and y_j\).

If two lines are perpendicular, then, the product of slopes equals -1. Hence,

$$a * m = -1$$

$$a = -1 / m$$

To find the coefficient b also known as the y-intercept, substitute the coordinates (x, y) and the value of “a” into the equation of your line:

$$y = ax + b$$

$$y = -1 . x / m + b$$

$$b = y+ 1 . x / m$$

You can verify the above results using our online perpendicular slope calculator.

Find Intersection Point of Perpendicular Lines:

To find the intersection points between two lines, first identify a point with coordinates \((x_j,y_j)\) which, lies on each of the two lines.

When we found two perpendicular lines: y = 3x – 6 and y = – 0.334x + 9.67. Then, these two equations form the equations with two unknowns.

Let’s solve these equations:

$$Y_j = 3x_j – 6$$

$$Y_j = -0.334x_j + 9.67$$

By multiplying the 9 with the second equation

$$Y_j = 3x_j – 6$$

$$9y_j = -3x_j + 87$$

Sum these two equations,

$$10y_j = 87$$

So,

$$y_j = 8.7$$

$$x_j = 0.5y_j + 1 = 6.96 + 1 = 7.96$$

Hence, the coordinates of the point of intersection are \((7.96, 8.7)\).

How to Find the Slope of a Parallel Line?

Take a simple line equation:

$$y = ax + b$$

Where x is the x-coordinate, y is the y-coordinate, “a” and “b” are coefficients.

And the coordinates of the point of the line are x and y.

If two lines are parallel with the same slop. Then, according to the perpendicular slope formula:

$$a = m$$

When you plug in the coordinates \((x_j,y_j)\) and also the value of a into the equation line.

$$y = ax + b$$

$$y_j = m . x_j + b$$

Furthermore, an Online Point Slope Form Calculator will find the equation of a line by using two coordinate points and the slope of the line.

Find Distance Between Two Parallel Lines:

First know the equation of your line, which, you can use to find the distance between it and the other line. However, an online parallel line calculator determines the distance between two parallel lines automatically. Otherwise, you can use this distance formula:

$$D = \frac{|b – r|} { \sqrt(m^2 + 1)}$$

How Parallel and Perpendicular Line Calculator Works?

This parallel line equation calculator provides the slopes of parallel and perpendicular lines with the following steps:

Input:

• First, select the line type from a drop-down list.
• Then, choose the parallel or perpendicular line which, your desire to compute.
• Now, plug in the values in a specific box of the parallel equation calculator.
• Click the “Calculate.”

Output:

• The parallel or perpendicular calculator first displays your entered values, answer, and interception points.
• Then, the parallel line calculator provides a parallel line equation and the equation of the line.

FAQ:

Define the rule of parallel lines?

Perpendicular slopes are –ve reciprocals of each other. In simple words, two lines are parallel when their slopes are equal and they have different y-intercept.

What is the slope of perpendicular?

If a line is perpendicular to a line that has slop j, then the slope is -1/j. For instance, the slope of line x = (1/2)y + 3 is 1/2. Also, a free online perpendicular slope calculator helps to find the perpendicular line of the graph.

Is a slope of Zero (0) linear?

A zero (0) slope means there is no change in the x-coordinates as the y-coordinates change, so the line is horizontal. However, you can check whether zero is linear or not with help of our online slope of parallel and perpendicular lines calculator.

Final Point:

Use this free online perpendicular line calculator that allows you to find the slope-intercept of the equation of a line and the equation of the parallel line. Additionally, this perpendicular line equation calculator provides the interception points where two lines meet with each other. This online determine whether the lines are parallel perpendicular or neither calculator is the best choice for everyone.

Reference: 

From the source of Wikipedia: Foot of a perpendicular, Construction of the perpendicular, In relationship to parallel lines, Graph of functions.

From the source of Splash Learn: What is Perpendicular, Properties of perpendicular lines, perpendicular lines.

From the source of Khan Academy: Equations of parallel and perpendicular lines, Types of Problems, Strategies, Real-life Applications.

How do you find the equation of a parallel line?