Equation of Line
The equation of line is an algebraic form of representing the set of points, which together form a line in a coordinate system. The numerous points which together form a line in the coordinate axis are represented as as (x, y) and the relation between x and y forms an algebraic equation, which is referred to as an equation of a line. Using the equation of any line, we can find whether a given point lies on the line or not.
The equation of line is a linear equation with a degree of one. Let us understand more about the different forms of the equation of a line and how to find the equation of line.
1.  What is Equation of a Line? 
2.  Standard Form of Equation of a Line 
3.  Equation of a Line Formula 
4.  How to Find Equation of Line? 
5.  FAQs on Equation of a Line 
What is Equation of a Line?
The equation of a line is linear in the variables x and y which represents the relation between the coordinates of every point (x, y) on the line. i.e., the equation of line is satisfied by all points on it.
The equation of a line can be formed with the help of the slope of the line and a point on the line. Let us understand more about the slope of the line and the needed point on the line, to better understand the formation of the equation of a line. The slope of the line is the inclination of the line with the positive xaxis and is expressed as a numeric integer, fraction, or the tangent of the angle it makes with the positive xaxis. The point refers to a point on the with the x coordinate and the y coordinate
The general form of the equation of a line with a slope m and passing through the point (x_{1}, y_{1}) is given as: y  y_{1} = m(x  x_{1}). Further, this equation can be solved and simplified into the standard form / slopeintercept form / intercept form of the equation of a line.
Standard Form of Equation of a Line
The standard form of equation of a line is ax + by + c = 0. Here a, b, are the coefficients, x, y are the variables, and c is the constant term. It is an equation of degree one, with variables x and y. The values of x and y represent the coordinates of the point on the line represented in the coordinate plane. The following quick rules are to be followed in the process of writing this standard form of the equation of a line.
 The x term is written first, followed by the yterm, and finally, the constant term is written.
 The coefficients and the constant values, should not be written as fractions or decimals and should be written as integers.
 The value of 'a', the coefficient of x is always written as a positive integer.
Equation of line in standard form: ax + by + c = 0
where,
 a, b are coefficients
 x, y are variables
 c is constant
Presented below are the five different forms of equations of a line. All of these can be transformed and presented in standard form.
Equation of a Line Formula
There are about five basic different formulas of writing the equation of line based on the parameters known for the line. These different formulas used to find and represent the equation of a line are as given below,
 Point Slope Form: (y  y_{1}) = m(x  x_{1})
 Two Point Form: (y y_{1}) =[(y_{2}  y_{1}) / (x_{2}  x_{1})] (x  x_{1})
 Slopeintercept Form: y = mx + c
 Intercept Form: x/a + y/b = 1
 Normal Form: x cos θ + y sin θ = p
Let us try and understand more about each one of these forms of the equation of a line.
Point Slope Form of Equation of Line
The pointslope form of the equation of a line requires a point on the line and the slope of the line. If (x_{1}, y_{1}) is a point on the line and the slope of the line is m, then the equation of a line in pointslope form is:
(y  y_{1}) = m(x  x_{1})
Here m = slope of the line and a line can have a positive slope, negative slope or a zero slope.
Two Point Form of Equation of Line
The twopoint form of the equation of a line is an extension of the pointslope form of the equation of a line. In the pointslope form of the equation of a line the slope m = (y_{2}  y_{1})/(x_{2}  x_{1}) is substituted to form the twopoint form of the equation of a line. The line equation from two points (x_{1}, y_{1}), and (x_{2}, y_{2}) is given by the twopoint form is as follows.
(y y_{1}) = [(y_{2}  y_{1}) / (x_{2}  x_{1})] (x  x_{1})
Slope Intercept Form of Equation of Line
The slopeintercept form of a line is y = mx + c. Here m is the slope of the line and 'c' is the yintercept of the line. This line cuts the yaxis at the point (0, c) and c is the distance of this point on the yaxis from the origin. The slopeintercept form of the equation of a line is important and has great applications in different topics of mathematics and engineering.
y = mx + c
Intercept Form of Equation of Line
The equation of a line in intercept form is formed with the xintercept 'a' and the yintercept 'b'. The line cuts the xaxis at the point (a, 0), and the yaxis at the point(0, b), and a, b are the respective distances of these points from the origin. Further, these two points can be substituted in the twopoint form of the equation of a line and simplified to get this intercept form of the equation of the line. The equation of line in intercept form is:
x/a + y/b = 1
Equation of a Line Using Normal Form
The normal form of the equation of a line is based on the perpendicular drawn to the line from the origin. The line perpendicular to the given line, and which passes through the origin is called the normal. Here the parameters of length of the normal 'p' and the angle made by the normal 'θ' with the positive xaxis are useful to form the equation of the line. The normal form of the equation of a line is as follows:
x cos θ + y sin θ = p
☛ Also check: Further, in addition to the abovedefined forms of the equation of a line, we can also use the equation of line calculator to conveniently find the equation of a line in quick and easy steps. Also, to use this equation of a line calculator, we need to provide the values of slope m and the yintercept c, to obtain the answer of the equation of a line in slopeintercept form and the standard form.
How to Find Equation of Line?
For finding the equation of a line, we can apply the formulas for any of the forms explained above, depending upon the data known to us. The steps that can be followed for different cases based on the known parameters and the form are as given below,
 Step 1: Note down the provided data, slope of line as 'm' and coordinates of the given point(s) in form (x_{n}, y_{n}).
 Step 2: Apply the required formula depending upon the given parameters,
(i) For finding the equation of a line, given its slope or gradient and its intercept on the yaxis, use slopeintercept form.
(ii) To find the equation of a line, given its slope and coordinates of one point that lies on the line, use the pointslope form.
(iii) For finding the equation of a line, given the coordinates of two points lying on it, use the twopoint form.
(iv) To write an equation, given the xintercept and yintercept, use the intercept form.  Step 3: Rearrange the terms to express the equation of the line in standard form.
Note: The alternate method for cases (ii), (iii), and (iv) can be to first calculate the slope by applying the slope formula using the given data and then finally applying the slopeintercept formula.
Equation of Horizontal and Vertical Line
We do not need any of the above formulas to find the equation of a horizontal/vertical line.
 The equation of a horizontal line (a line parallel to the xaxis) is found using the general equation: y = b, where b is the ycoordinate of any point lying on the line.
 Similarly, the equation of a vertical line (a line parallel to the yaxis) can be given as: x = a, where, a is the xcoordinate of any point lying on the given line.
Using the same rules, one can see that the equation of the xaxis is y = 0 and the equation of the yaxis is x = 0.
☛ Related Topics:
Important Notes on Equation of Line:
 The equation of xaxis is y = 0 and the equation of yaxis is x = 0.
 The equation of a line parallel to the xaxis is y = b, where it cuts the yaxis at the point (0, b).
 The equation of a line parallel to the yaxis is x = a, where it cuts the xaxis at the point (a, 0).
 The equation of a line parallel to ax + by + c = 0 is ax + by + k = 0.
 The equation of a line perpendicular to ax + by + c = 0 is bx  ay + k = 0.
Examples on Equation of Line

Example 1: Derive the normal form of the equation of a line.
Solution:
Let the length of the normal be P and it is inclined at an angle θ with the positive xaxis.
The projection of the normal on the xaxis and yaxis is Pcosθ and Psinθ respectively.
The coordinates of the point P is (Pcosθ, Psinθ).
The slope of the normal is tanθ, and the slope of the required line which is perpendicular to the normal is 1/tanθ
Now we have the point (Pcosθ, Psinθ), and the required slope m = 1/Tanθ to form the equation of the line.
(y  Psinθ) = 1/tanθ. (x  Pcosθ)
(y  Psinθ) = 1/sinθ/cosθ. (x  Pcosθ)
(y  Psinθ) = cosθ/sinθ. (x  Pcosθ)
sinθ(y  Psinθ) = cosθ. (x  Pcosθ)
ysinθ  Psin^{2}θ = xcosθ +Pcos^{2}θ
xcosθ + ysinθ = Psin^{2}θ + Pcos^{2}θ
xcosθ + ysinθ = P(sin^{2}θ + cos^{2}θ)
xcosθ + ysinθ = P
☛Also check: Equation of a Line Worksheets
Answer: Hence, the normal equation of a line is proved.

Example 2: Find the equation of a line having an xintercept of 5 units and a yintercept of 4 units. Also, represent this equation in standard form.
Solution:
The given xintercept is a = 5, and y = 4.
Applying this in the intercept form of the equation of a line x/a + y/b = 1, we have the equation of a line as follows.
x/5 + y/4 = 1
Further, we need to convert this equation into standard form.
x/5 + y/4 = 1
(4x + 5y)/20 = 1
4x + 5y = 20
4x + 5y  20 = 0
Answer: Hence the standard form of the equation of a line is 4x + 5y = 20.

Example 3: Find the slope and yintercept of the line with equation 3x  4y + 7 = 0.
Solution:
The given equation of the line is 3x  4y + 7 = 0
To find the yintercept and the slope from equation of a line, we need to convert this equation into slopeintercept form.
3x  4y + 7 = 0
3x + 7 = 4y
4y = 3x + 7
y = 3x/4 + 7/4
Comparing this equation with the slopeintercept form of the equation of line y = mx + c we have the slope m = 3/4, and the yintercept c = 7/4.
Answer: Slope m = 3/4, and yintercept c = 7/4.
FAQs on Equation of a Line
What is the Equation of a Line in Coordinate Geometry?
The equation of a line is a single representation of numerous points on the line. The general form of the equation of a line is of the form ax + by + c = 0 and any point on the line satisfies this equation. Most of the time the equation of a line with slope 'm' and a point (x_{1}, y_{1}) on it is found by the formula y  y_{1} = m × (x  x_{1}).
What is the Process of Writing Equations of Lines?
For writing equations of lines, first note down the known information about it. Usually, some information like slope, point(s) on it, its intercepts, the angle made by the line with xaxis, etc will be given in the problems. Then we can use one of the equation of line formulas mentioned in the page above here.
What is the Equation Of a Line Parallel To The XAxis?
The equation of a line parallel to the xaxis is of the form y = b, which cuts the yaxis at the point (0, b). An example is the equation of the line y = 5, which is parallel to the xaxis and cuts the yaxis at the point (0. 5). Also, the points such as (2, 5), (3, 5), etc. lie on this line y = 5 as their ycoordinate as 5.
What is the Equation of a Line in SlopeIntercept Form?
The slopeintercept form of the equation of a line is y = mx + c, where m is the slope of the line, and c is the yintercept of the line.
 The slope of this line 'm' is a numeric value that indicates the inclination of the line, and is also equal to the tan of the angle the line makes with the positive xaxis.
 The yintercept 'c' is is the distance of the point on the yaxis where this line cuts the yaxis from the origin.
What is the Equation of a Line Passing Through Two Points?
The equation of a line in twopoint form is (y  y_{1}) = (y_{2}  y_{1})/(x_{2}  x_{1}) . (x  x_{1}). Here (y_{2}  y_{1})/(x_{2}  x_{1}) is the slope of the line and this line is passing through the two points (x_{1}, y_{1}), and (x_{2}, y_{2}). This twopoint form is an interpretation of the pointslope form.
What is the Equation of a Line in Standard Form?
The standard form of equation of a line is ax + by + c = 0. Here a, b, are the coefficients, x, y are the variables and c is the constant term. Any ordered pair (x, y) lying on the line satisfies this equation.
What is the Equation of a Line Perpendicular to Another Line?
The equation of a line drawn perpendicular to the line ax + by + c = 0 is bx  ay + c = 0. Let us understand this with a quick example. The equation of line perpendicular to the line 4x + 3y + 7 = 0 is 3x  4y + k = 0. Here, k is the constant and its value can be obtained by substituting any point in this equation of this line.
How To Find the Slope of a Line Equation?
The slope of a line equation is ax + by + c = 0 is  a/b. Also, the given equation of a line can be converted from standard form to slope intercept form, and the coefficient of the x would be the slope of the line. As an example we can obtain the slope of a line having an equation of a line 4x  5y + 11 = 0 by using the formula to obtain the slope as (4/5) = 4/5.
How To Find the Equation of a Line With One Point?
The equation of a line with one given point (x_{1}, y_{1}) is (y  y_{1}) = m(x  x_{1}). Here m is the slope of the line. Further, this equation can finally be solved and presented in the standard form as ax + by + c = 0. Let us find the equation of a line passing through the point (2, 1) and having a slope of 3. The required equation of the line using this one point form is (y  1) = 3(x  2), which on simplification gives the final equation in standard form as 3x  y  5 = 0
How To Find the Equation of a Line Parallel to a Line?
The equation of a line parallel to the given line would be the same, but the constant term would be different. The equation of a line parallel to the line ax + by + c = 0 would be ax + by + k = 0. Here k is a constant term that can be obtained by substituting any point lying on the line, in the equation of the line. Example: The equation of a line parallel to the line 5x + 6y + 11 = 0 is 5x + 6y + k = 0.
How to Find the Equation of a Line When the Slope is Undefined?
The line whose slope is not defined is either the yaxis or a line parallel to the yaxis. Hence the equation of a line whose slope is not defined is x = a, and it cuts the xaxis at the point (a, 0).
What is Vector Equation of a Line?
The vector equation of a line passing through a point (with position vector a) and parallel to a vector b is r = a + λb, here by substituting different values of λ, we can get different points on the line.
How to Find the Equation of a Line When the Graph of the Line Is Given?
The equation of a line from the graph of the line can be easily obtained by taking two points falling on the line of the graph. Further, the two points can be used, and with the help of the twopoint form of the equation of a line, we can find the required equation.
What is c in the SlopeIntercept Form Of The Equation Of a Line?
The 'c' in the slopeintercept form of the equation of a line y = mx + c is the yintercept of the line. The line cuts the yaxis at the point (0, c), and c is the distance of the point on the yaxis from the origin. Different notations are adopted to represent the equation of line in different countries. Some common notations are 'y = mx + b', 'y = mx + c', 'y = ax + b', etc.
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