Line Calculator

This calculator will help you to find Equation of a line joining two points P_1 (x_1, y_1) and P_2 (x_2, y_2) with the steps shown.
Related Calculators:Line intersection in 2-D Calculator

Your Input :-

Your input can be in form of FRACTION, Real Number or any Variable

Point P1:

1.4

Point P2:

Formatted User Input Display

\sf{ \space \space \space \space \space \space \space \space \space \space (x_1, \space y_1) \space = \space \bold{({1.4}, \space {5})} \space \space \space \space \space \space \space \space }

\sf{ \space \space \space \space \space \space \space \space \space \space (x_2, \space y_2) \space = \space \bold{({7}, \space {2})} \space \space \space \space \space \space \space \space }

Step By Step Solution :-

\sf{The \space \space Intercept \space \space form \space \space of \space \space a \space \space line \space \space passing \space \space through \space two \space \space points \space \space P= \space \bold{(1.4, \space 5)} \space \space and \space \space Q \space = \space \bold{(7, \space 2)}}

\sf{is \space \space given \space \space by \space \space the \space \space following \space \space formula \space : \space {x\above{1pt}a} \space + \space {y\above{1pt}b} \space = \space 1}

\sf{where \space \boldsymbol{a} \space is \space the \space intercept \space on \space \boldsymbol{x} \space - \space axis \space \enspace \space and \space \space \boldsymbol{b} \space \space is \space \space the \space \space intercept \space \space \space \space \space \space \space \space on \space \space \boldsymbol{y} \space - \space axis}

\sf{so \space \space a \space = \space (x_1 \space - \space {y_1\above{1pt}m}) \space \space and \space \space b \space = \space (y_1 \space - \space mx_1 \space )}

\sf{Given \space the \space values \space of \space the \space variables \space :}

\sf{x_1 \space = \space \bold{\bold{{1.4} \space \space or \space \space {7\above{1pt}5}}}, \space y_1 \space = \space \bold{{5}}, \space x_2 \space = \space \bold{{7}}, \space y_2 \space = \space \bold{{2}}}

\sf{After \space \space putting \space \space the \space \space values \space \space in \space \space the \space \space formula}

Lets calculate the value of m

\sf{m = {(y_2 - y_1)\above{1pt}(x_2 - x_1)}}

\sf{m = {(2 - 5)\above{1pt}(7 - 1.4)}}

\sf{m = {{-3}\above{1pt}{28\above{1pt}5}} \implies m = {{-15}\above{1pt}{28}}}

Now we have to calculate the value of a

\sf{a = (x_1 - {y_1\above{1pt}m}) \space or \space a' = (x_2 - {y_2\above{1pt}m})}

\sf{result \space \space will \space \space be \space \space same \space \space in \space \space both \space \space cases}

After putting the values in formula

\sf{a = (1.4 - {5\above{1pt}{{-15}\above{1pt}{28}}}) \implies a = {161\above{1pt}15}}

\sf{a' = (7 - {2\above{1pt}{{-15}\above{1pt}{28}}}) \implies a' = {161\above{1pt}15}}

\sf{Hence \space value \space of \space a = {161\above{1pt}15}}

Now we have to calculate the value of b

\sf{b = (y_1 - mx_1) \space or \space b' = (y_2 - mx_2)}

\sf{result \space \space will \space \space be \space \space same \space \space in \space \space both \space \space cases}

After putting the values in formula

\sf{b = ({5} - ({{-15}\above{1pt}{28}}*1.4)) \implies b = {23\above{1pt}4}}

\sf{b' = (2 - ({{-15}\above{1pt}{28}}*{7})) \implies b' = {23\above{1pt}4}}

\sf{Hence \space value \space of \space b = {23\above{1pt}4}}

Now we can write the intercept form equation of line as given below :-

\sf{{x\above{1pt}{161\above{1pt}15}} + {y\above{1pt}{23\above{1pt}4}} = 1}

Final Answer

\sf{The \space \space Intercept \space \space form \space \space of \space \space a \space \space line \space \space passing \space \space through \space two \space \space points \space \space P=\bold{(1.4, \space 5)} \space \space and \space \space Q=\bold{(7,2)} \space \space is \space : \space }

\sf{\bold{{x\above{1pt}{161\above{1pt}15}} + {y\above{1pt}{23\above{1pt}4}} = 1}}

Note :- If you find any computational or Logical error in this calculator, then you can write your suggestion by clicking the below button or in the comment box.

$\color{red} \bold{Related \space Calculators:}$
Slope of a line
Section formula
Angle between two lines
Distance between two points
Parallel and Perpendicular line
Concurrency of Straight lines
Foot of perpendicular to a given line

$\bold{Table \space of \space Content}$

1. Introduction to the finding the equation of a line from two points in 2D
2. What is the Formulae used ?
3. How do I calculate the equation of a line from two points in 2D?
4. Why choose the equation of a line from two points in 2D Calculator?
5. A Video for explaining this concept
6. How to use this calculator ?
7. Solved Examples
8. Frequently Asked Questions (FAQs)
9. What are the real-life applications?
10. Conclusion

1. Introduction to the finding the equation of a line from two points in 2D

Understanding how to find the equation of a line from two given points is a fundamental skill in algebra and geometry. In this comprehensive guide, we'll explore the various forms of linear equations, including intercept form, general standard form, point-slope form, and slope-intercept form. By the end of this article, you'll be equipped with the knowledge to effortlessly transform between these forms and apply them to real-world scenarios.
$\bold{Definition}$
A linear equation represents a straight line on a graph and can be expressed in different forms. The general form of a linear equation is Ax + By = C, where A, B, and C are constants. We'll delve into specific forms and techniques to find the equation of a line from two given points.

2. What is the Formulae used?

There are various forms of lines exist that are discussed below:
$\color{black}\bold{Intercept \space form \space of \space line: }$
$(\frac{x}{a}) + (\frac{y}{b}) = 1$ , where $\color{black}\bold{a}$ is the intercept on x - axis and $\color{black}\bold{b}$ is the intercept on y - axis.
$\color{black}\bold{General \space standard \space form \space of \space line:}$ ${Ax + By = C}$ .
$\color{black}\bold{Point \space slope \space form \space of \space line: }$
$(y - y_1) = m(x - x_1)$ , where $\color{black}\bold{m}$ is the slope of the line and $\bold{(x_1, y_1)}$ is the point lying on the line.
$\color{black}\bold{Slope \space intercept \space form \space of \space line: }$
$y = mx + c$ , where $\color{black}\bold{m}$ is the slope of the line and $\bold{c}$ is the intercept on y - axis.

3. How do I calculate the equation of a line from two points in 2D?

Calculating the equation of a line from two points in 2D involves a series of straightforward steps:
Identify the given point's coordinates.
Plug these values into the input boxes.
Select the equation type.
See the step-wise result by clicking on the calculate button.

4. Why choose our equation of a line from two points in 2D Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on the concept of how to find the equation of a line from two points in 2D.

6. How to use this calculator

This calculator will help you find a line's equation from two points in 2D.
In the given input boxes, you have to put the value of the coordinates.
After clicking the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question:1}$
Find the equation of the line passing through the points $P_1 (5, 2)$ & $P_2 (3, 7)$ .
$\bold{Solution:1}$
$\bold{Step \space 1:}$ First we will find the slope m of the line $m = \frac{7-2}{3-5} = \frac{-5}{2}$
$\bold{Step \space 2:}$ we can write the equation of line as $(y - 2) = \frac{-5}{2} (x - 5)$ And now, we can convert the above equation to the desired form.

$\bold{Question:2}$
Find the equation of the line passing through the points $P_1 (-3, 0)$ & $P_2 (-7, 5)$ .
$\bold{Solution:2}$
$\bold{Step \space 1:}$ First we will find the slope m of the line $m = \frac{5-0}{-7+3} = \frac{-5}{4}$
$\bold{Step \space 2:}$ we can write the equation of line as $(y - 0) = \frac{-5}{4} (x + 3)$
Now we can convert the above equation to the desired form.

8. Frequently Asked Questions (FAQs)

What is the significance of finding the equation of a line from two points in different forms?

Expressing the equation of a line in various forms provides flexibility in mathematical analysis. Different forms can be more suitable for specific applications, making it easier to interpret and manipulate equations based on the problem.

How does the choice of form impact the ease of solving real-world problems using linear equations?

Each form of a linear equation offers a unique perspective on the relationship between variables. Choosing the appropriate form can simplify calculations, aid in graphical representation, and enhance the understanding of specific characteristics, making problem-solving more efficient.

Can the equation of a line be expressed in all forms simultaneously, or is it necessary to choose one form over another?

While it is possible to convert between different forms, expressing the equation in all forms simultaneously is uncommon. Typically, choosing a specific form depends on the problem's requirements or the ease of interpretation for a given context.

In what real-life scenarios is the ability to switch between different forms of linear equations most beneficial?

Professionals in fields like physics, engineering, and economics often encounter scenarios where different forms of linear equations are useful. For instance, engineers may prefer the slope-intercept form for designing structures, while economists might find the point-slope form more intuitive for analyzing economic trends.

Are there any shortcuts or rules of thumb for quickly identifying which form of a linear equation is most appropriate for a given problem?

While there's no one-size-fits-all rule, certain considerations can guide your choice. For example, the slope-intercept form is convenient if you know the slope and y-intercept. If you have specific points, the point-slope form might be efficient. Understanding the problem context and available information will help you make an informed decision.

9. What are the real-life applications?

Understanding and applying these forms of linear equations are crucial in various real-world scenarios. Architects use them to design structures, economists model economic trends, and engineers design electrical circuits. Expressing and manipulating equations in different forms enhances problem-solving skills across diverse fields.

10. Conclusion

Mastering the art of finding the equation of a line from two given points in various forms opens up a world of mathematical possibilities. Whether solving complex engineering problems or analyzing economic data, the flexibility to express linear equations in different forms is a valuable skill. Keep practicing and applying these concepts to build a solid algebra and mathematical problem-solving foundation.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: doubt@doubtlet.com

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