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.

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 x-axis and is expressed as a numeric integer, fraction, or the tangent of the angle it makes with the positive x-axis. 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 (x₁, y₁) is given as: y - y₁ = m(x - x₁). Further, this equation can be solved and simplified into the standard form / slope-intercept form / intercept form of the 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 y-term, 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.

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 - y₁) = m(x - x₁)
• Two Point Form: (y -y₁) =[(y₂ - y₁) / (x₂ - x₁)] (x - x₁)
• Slope-intercept 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 point-slope form of the equation of a line requires a point on the line and the slope of the line. If (x₁, y₁) is a point on the line and the slope of the line is m, then the equation of a line in point-slope form is:

(y - y₁) = m(x - x₁)

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 two-point form of the equation of a line is an extension of the point-slope form of the equation of a line. In the point-slope form of the equation of a line the slope m = (y₂ - y₁)/(x₂ - x₁) is substituted to form the two-point form of the equation of a line. The line equation from two points (x₁, y₁), and (x₂, y₂) is given by the two-point form is as follows.

(y -y₁) = [(y₂ - y₁) / (x₂ - x₁)] (x - x₁)

Slope Intercept Form of Equation of Line

The slope-intercept form of a line is y = mx + c. Here m is the slope of the line and 'c' is the y-intercept of the line. This line cuts the y-axis at the point (0, c) and c is the distance of this point on the y-axis from the origin. The slope-intercept 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 x-intercept 'a' and the y-intercept 'b'. The line cuts the x-axis at the point (a, 0), and the y-axis 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 two-point 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 x-axis 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 above-defined 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 y-intercept c, to obtain the answer of the equation of a line in slope-intercept form and the standard form.

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 y-axis, use slope-intercept form.
  (ii) To find the equation of a line, given its slope and coordinates of one point that lies on the line, use the point-slope form.
  (iii) For finding the equation of a line, given the coordinates of two points lying on it, use the two-point form.
  (iv) To write an equation, given the x-intercept and y-intercept, 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 slope-intercept 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 x-axis) is found using the general equation: y = b, where b is the y-coordinate of any point lying on the line.
• Similarly, the equation of a vertical line (a line parallel to the y-axis) can be given as: x = a, where, a is the x-coordinate of any point lying on the given line.

Using the same rules, one can see that the equation of the x-axis is y = 0 and the equation of the y-axis is x = 0.

☛ Related Topics:

• Coordinate Plane
• Equation of Circle
• Cartesian Form
• Equation of Plane

Important Notes on Equation of Line:

• The equation of x-axis is y = 0 and the equation of y-axis is x = 0.
• The equation of a line parallel to the x-axis is y = b, where it cuts the y-axis at the point (0, b).
• The equation of a line parallel to the y-axis is x = a, where it cuts the x-axis 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.

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 (x₁, y₁) on it is found by the formula y - y₁ = m × (x - x₁).

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 x-axis, 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 X-Axis?

The equation of a line parallel to the x-axis is of the form y = b, which cuts the y-axis at the point (0, b). An example is the equation of the line y = 5, which is parallel to the x-axis and cuts the y-axis at the point (0. 5). Also, the points such as (2, 5), (-3, 5), etc. lie on this line y = 5 as their y-coordinate as 5.

What is the Equation of a Line in Slope-Intercept Form?

The slope-intercept form of the equation of a line is y = mx + c, where m is the slope of the line, and c is the y-intercept 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 x-axis.
• The y-intercept 'c' is is the distance of the point on the y-axis where this line cuts the y-axis from the origin.

What is the Equation of a Line Passing Through Two Points?

The equation of a line in two-point form is (y - y₁) = (y₂ - y₁)/(x₂ - x₁) . (x - x₁). Here (y₂ - y₁)/(x₂ - x₁) is the slope of the line and this line is passing through the two points (x₁, y₁), and (x₂, y₂). This two-point form is an interpretation of the point-slope 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 (x₁, y₁) is (y - y₁) = m(x - x₁). 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 y-axis or a line parallel to the y-axis. Hence the equation of a line whose slope is not defined is x = a, and it cuts the x-axis 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 two-point form of the equation of a line, we can find the required equation.

What is c in the Slope-Intercept Form Of The Equation Of a Line?

The 'c' in the slope-intercept form of the equation of a line y = mx + c is the y-intercept of the line. The line cuts the y-axis at the point (0, c), and c is the distance of the point on the y-axis 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.