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Equation of the Line: Definition and Examples

write the equation of the line

In two-dimensional geometry, a coordinate plane has infinite points. Suppose you are given a point A (x,y) on the XY plane. You are also given a line L. Now, how would you determine whether the given point A lies on the given line L? Here is when you should know how to write the equation of the line and how to check if the given point satisfies the equation. If the given point satisfies the equation of the line, then it lies on the line. Read on to learn more about how to write the equation of a line.

Here’s what we’ll cover in the article:

  • What is the equation of a straight line?
  • What are the different forms of the equation of a line?
  • How do you write the equation of a line?
  • How to write an equation of the line that passes through a point?
  • How to write the equation of a line that is perpendicular?
  • How to write the equation of a tangent line?

What is the equation of a straight line?

A straight line is obtained when two points are connected with a minimum distance. An equation of a line is a simple mathematical equation that provides the relation between the points lying on that line via a slope, x-intercept, and y-intercept of the line. You can write the equation of the line in different forms. The most common form of the equation of a straight line is as follows:

y = mx + b  

In the above equation, 

y tell how far up

x is how far along

m is the gradient or slope, i.e., how steep

b is the value of y when x is 0

If you are given a graph, how would you find the values of the variables?

b is the point where the line intersects the y-axis.

m = change in y /change in x

What are the different forms of the equation of a line?

Following are the different forms of the equation of a line.  

Standard Form                           ax + by = c 
Point Slope Form             y – y1 = m (x – x1)
Slope intercept form                  y = mx + b
Two point form             ( y-y1 ) = (y2 – y1)(x2 -x1) (x -x1)
Intercept form                          xa+yb=1 Normal form                         xcosθ + ysinθ = P

How do you write the equation of a line in standard form?

The equation of line in standard form is written as ax + by + c = 0. 

Here. 

a and b are the coefficients

x and y are the variables (representing the coordinates of the point on the line)

c is the constant term

You must keep the following rules in mind while writing the standard form of the equation of a line:

  • Begin by writing the x-term followed by the y-term. The constant term is written at the end.
  • The constant values and the coefficients are written as integers, not fractions or decimals.  
  • The coefficient of x, the value of ‘a’, is always written as a positive integer.

Equation of the line in point-slope form 

The point-slope form requires the slope of the line and a point on the line. The point on the line is (x1, y1) while the slope is m. The numeric values of the point represent the x coordinate and the ‘y’ coordinate. The slope of the line is its inclination with the positive x-axis. The slope can be positive, negative, or zero. Thus, you write the equation of the line as follows.

(y – y1) = m(x – x1)

Equation of a line in the two-point form

The two-point form is a further explanation of the point-slope form. Here the slope of line m is substituted by (y2- y1)/(x2 – x1) to form the two-point form. Thus, the equation of a line that passes through the two points (x1, y1) and (x2, y2) can be represented as follows: 

  ( y-y1 ) = (y2 – y1)(x2 -x1) (x -x1)

Equation of a line in slope-intercept form 

The slope-intercept form is the most commonly used form of the equation of a line. It is written as:

y = mx + b

Where m is the slope of the line, and ‘b’ represents the y-intercept of the line. The y-intercept is obtained when the line cuts the y-axis at the point (0, b), and b gives the distance of this point from the origin on the y-axis.  

Equation of a line in intercept form 

When you are given the x-intercept ‘a’ and the y-intercept ‘b,’ i.e., the line cuts the x-axis at the point (a, 0), and the same line cuts the y-axis at the point (0, b). Thus, a and b are the respective distances of these points from the origin. When substituted in the two-point form of the equation of a line, these two points give the intercept form of the equation of the line.

The intercept form is written as  xa+yb=1

Equation of a line in normal form

You can write the equation of the line in normal form when you know the perpendicular that passes through the origin. The perpendicular line to the given line and passes through the origin is called the normal. 

Here, p is the normal length, and θ is the angle made by the normal with the positive x-axis. The normal form is written as follows: 

xcosθ + ysinθ = P

How to write an equation of the line that passes through a point?

We can find the equation of a line by applying the formulas for any of the above-given forms. The form will depend on the data known to us. The steps that you must follow are as follows: 

Step 1: Write down the provided data, slope as ‘m’, and coordinates of a given point(s) in the form (xn, yn).

Step 2: Use the required formula to write the equation, depending upon the given parameters.

(i) If you are given the slope or gradient of a straight line and its intercept on the y-axis, use the slope-intercept form of the equation of the line.

(ii) If given the slope and coordinates of a point on the line, use the point-slope form.

(iii) If given the coordinates of two points, use the two-point form.

(iv) If given the x-intercept and y-intercept, use the intercept form.

Alternatively, you can also begin by calculating the slope by using the slope formula and then applying the slope-intercept formula.

How to write the equation of a line that is perpendicular to a line?

When given two lines that are perpendicular to each other, i.e., they meet at an angle of 90 degrees; you can find the slope as follows:

Suppose the slope of a line is m. The slope of the perpendicular line will be (-1/m). So, the slope of the line perpendicular to the given is always its negative reciprocal. 

Also, two lines are perpendicular to each other if the product of their slopes is -1. 

The following example will help you understand the concept of writing the equation of a line that is perpendicular to a line.

Example: Write the equation of the line that is perpendicular to 3x+5y=15 and passes through (3,2)?Solution: Firstly, we will write the equation, 3x+5y=15, in slope-intercept form.5y=−3x+15y = -35+3From the equation, we can state that the slope is -3/5So, the slope of the line perpendicular to this line will be its negative reciprocal, i.e., The slope = 5/3Now, using our new slope and the point (3,2) given in the question, we can find the y-intercept from the point-slope form of the equation of line as follows:2 = 53(3)+b2 = 5 + bb= -3Thus, the equation of the line can be written as:y = 53x-3

How to write the equation of horizontal and vertical lines?

A horizontal line that runs parallel to the x-axis has a slope of 0. Thus, we will substitute the values using the slope-intercept form of the equation of a line, y = mx + b. 

Since, m = 0, mx will also become 0.

So, we write the equation of the line as y = b. Here, b represents the y-coordinate of the y-intercept.

Similarly, a vertical line runs parallel to the y-axis and has a slope of 0. 

Here, the equation of the line will be written as y = b, where b is the x-coordinate of any point that lies on the given line.

Also, if you are given two straight lines that run parallel to each other, then they will have the same slope.

How to write the equation of a tangent line?

We use the point-slope form: y – y₀ = m (x – x₀) to write the equation of a line with slope ‘m’ that passes through a point (x₀, y₀). 

Now, if we consider a tangent line drawn to a curve: 

y = f(x) at a point (x₀, y₀). 

The slope of the tangent line = (f ‘(x)) ₍ₓ₀, ᵧ₀₎ 

Now substituting m, x₀, and y₀ in the point-slope form of the equation of the line, y – y₀ = m (x – x₀) 

The tangent line formula will be y – y₀ = (f ‘(x)) ₍ₓ₀, ᵧ₀₎ (x – x₀)

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