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 a set of variables x, y to form 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 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 in the coordinate system 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 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 are transformed and presented as a standard form.

There are about five basic different forms of writing the equation of line based on the parameters known for the straight line. These different forms used to find and represent the equation of a line are as given below,

• Point Slope Form
• Two Point Form
• Slope-intercept form
• Intercept form
• Normal form

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. The referred point on the line is (x\(_1\), y\(_1\)) and the slope of the line is m. The point is a numeric value and representing the x coordinate and the y coordinate of the point and the slope of the line m is the inclination of the line with the positive x-axis. Here m can have a positive slope, negative slope or a zero slope. Hence of the equation of a line is as follows.

(y - y\(_1\)) = m(x - x\(_1\))

Two Point Form of Equation of Line

The two-point form of the equation of a line is a further explanation 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 = (y\(_2\) - y\(_1\))/(x\(_2\) - x\(_1\)) is substituted to form the two-point form of the equation of a line. The equation of a line passing through the two points (x\(_1\), y\(_1\)), and (x\(_2\), y\(_2\)) is as follows.

\[(y -y_1) = \frac{(y_2 - y_1)}{(x_2 - x_1)}(x - x_1) \]

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 an important form 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. This intercept form explains the distance at which the line cuts the x-axis and the y-axis from the origin.

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

Equation of a Line Using Normal Form

The normal form of the equation of a line is based on the perpendicular of line, which passes through 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 this 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,

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

☛ Related Topics:

• Slope Intercept Form of a Line
• 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, as it cuts the y-axis at the point (0, b).
• The equation of a line parallel to the y-axis is x = a, and 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?

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. These two minimum requirements to form the equation of a line are the slope of the line and a point on the line.

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) are all the points lying on this line y = 5 has 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 points on the y-axis from the origin, where this line cuts the y-axis.

What is the Equation of a Line With 2 Points?

The equation of a line in two-point 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 lies is passing through the two points (x\(_1\), y\(_1\)), and (x\(_2\), y\(_2\)). This two-point form of the equation of a line is an interpretation of the point-slope form of the equation of a line.

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. The other different forms to find and represent the equation of a line are slope-intercept form, point-slope form, two-point form, intercept form, and normal form.

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 Using the Equation of a Line?

The slope of a line having an equation is ax + by + c = 0 is - a/b. Also, the given equation of a line can be converted into slope-intercept form of an equation of a line, and the coefficient of the x-axis 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 is finally 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. 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 will have a slope m = 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).

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 (c, 0), 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.