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Equation of a Straight Line

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Introduction to Equation of a Straight Line

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In this article, we’ll discuss the equation of a straight line formula, gradient of a straight line formula. We’ll also discuss some examples of how to find the equation of a straight line when Co-ordinates or gradients are given to us. There are n-number of ways to express an equation of a straight line and some are more general than others.

In order to master the techniques explained here, it's vital that you simply undertake many practice exercises so that they become second nature.

After reading this article, you will be able to solve:

  • How to find the equation of a line, given its gradient and its intercept on the y-axis;

  • How to find the equation of a straight line, given its gradient and one point lying on it;

  • How to find the equation of a line given two points lying on it;

  • Find the equation of a straight line in either of the forms y = mx + c or ax + by + c = 0.


Define Equation of a Straight Line

So, how do we define the equation of a straight line? When we started studying geometry first we studied point and then line. The very first intuition is, a line is a set of points which holds good in certain conditions. The definition of a line is, a straight line is the collection of points in between and extending beyond two points. In most geometrical perspectives, a line is an object that does not have formal properties other than it’s length, it’s single-dimensional. You may find the various equations of a straight line calculator on the internet.



General Equation of a Straight Line Formula

The general equation of straight line is 

ax + by + c = 0

Here x and y are the coordinate axes and a, b ,c are the constants.


The Slope of a Straight Line

The slope of a straight line is also known as the gradient of a straight line. Actually, it is the tangent of an angle. An angle of the straight line from the positive direction of the x-axis. Usually, it is represented by m and its formula is m = tanፀ. Where ፀ (theta) is the angle of the straight line from the positive direction of the x-axis considered anticlockwise as mentioned in the following diagram.



There is one more formula for getting the slope of a line when two points are given, the line is passing through. Let (x1,y1) and (x2,y2) are the point and we are required to find the gradient then we’ll use the formula

\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]


Slope Intercept Form of the Straight Line

It is the most used and easiest form of the straight line. The equation for slope-intercept form is y = mx + c, where x and y are axes, m is the slope whose formula is tanፀ and c is the intercept of line on the y-axis.



The Intercept Form of the Straight Line

The intercept form of a straight line can be written as 

\[\frac{x}{a}+\frac{y}{b}=1\]

Here, a and y are coordinate axes and a, b are intercepted on the x and y-axis respectively.

The geometrical representation of intercept form is mentioned below:



The Normal Form of a Straight Line

The normal form of the straight line is 

x cos α + y sin α = p

Here, x and y are coordinates, p is the length of the perpendicular from origin to the straight line and α is the angle between the positive x-axis and the perpendicular of the straight line from the origin. As mentioned below



One important point we need to keep in mind is p is always positive and measured away from the origin. α is a positive angle that is less than 360⁰ measured from the positive direction of the x-axis, Many students misunderstand this. 


Slope One Point Form of the Straight Line

When the slope and a point through which the required straight line is passing, are given to us and we need to find the equation of the straight line the use the formula 

y - y1 = m(x - x1)

Here, (x1,y1) is the point on the coordinate plane through which the line is passing and m is the slope.



Two-Point Form of the Straight Line

In the previous section, we have discussed the one point form of the straight line which is y - y1 = m(x - x1)

In this formula, m is the slope and we know that when two points are given then the slope can be computed as \[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] . We can substitute this slope formula in the previous equation which will give us 

\[y-y_{1}=(\frac{y_{2}-y_{1}}{x_{2}-x_{1}})(x-x_{1})\]

Or, (y - y1)(y2 - y1) = (x - x1)(x2 - x1)

FAQ (Frequently Asked Questions)

1. How to Deduce Slope-Intercept Form From General Form?

Answer:  We know the general form of the straight line is ax + by + c = 0 To convert it into Slope intercept form we need to leave y alone on the left-hand side. That is

by = -ax-c

Now divide the whole equation by b. This will give us

y = -a/bx - c/b

This is the required slope-intercept form of the line. Where -a/b is the slope and -c/b is the intercept.

2. How to Deduce Intercept Form From General Form?

Answer:  To deduce intercept form from general from, we need to keep 1 alone on the right-hand side in the general equation which is ax + by + c = 0

ax + by = -c

Now to get 1 on the right-hand side, we’ll divide by -c into the whole equation. Which will give us

a/(-c)x + b/(-c)y = 1

This is the required intercept form of straight-line deduced from the general form. Here a/(-c) and b/(-c) are the intercept on the x and y-axis respectively.