How to calculate the equation of the line? Explained with examples

In mathematics, the equation of the line or linear equation of the straight line can be evaluated with the help of two well-known methods such as point-slope form and slope-intercept form. These methods or techniques are used on a wider scale.

There are different equations and calculations for these techniques for calculating the equation of the straight line. In this lesson, we will cover the basics of calculating the equation of the straight line with the help of slope intercept form and point-slope form.

What is the equation of the line?

In algebra, a line that is formed by taking the points of the line and the slope of the line is known as the linear equation of the line (equation of the straight line). It is a mathematical equation that offers the relation among the coordinate points (x-axis & y-axis) of the straight line.

For calculating the equation of the line, the basic knowledge of finding the slope of the line must be known. The slope of the line is an inclination or steepness of the line that is measured by dividing the change in the values of y and the change in the values of x.

Techniques of calculating the equation of the line

There are various techniques for evaluating the equation of a straight line. In this section, we are going to explain the two well-known techniques for finding linear equations.

• Slope intercept form
• Point slope form

Let us describe these techniques briefly.

• Slope intercept form

It is the most frequently used technique for finding the straight line equation of the line. This technique used a general equation to evaluate the equation of the line. The equation of the slope-intercept form is a combination of the fixed points of the line, the slope of the line, and the y-intercept form of the line.

The general equation of the slope-intercept form is:

y = m * x + b

• m = the slope (inclination or steepness) of the line.
• b = the y-intercept (y value) of the line.
• x & y = the fixed points of the line.

The equation of the line can be calculated with the help of the slope intercept form equation easily by following the below steps.

• First of all, calculate the slope of the line by using the coordinate points.
• After that calculate the y-intercept of the line.
• Substitute the value of slope and y-intercept of the line to the formula of the slope-intercept form.
• The generated equation will be the linear equation of the line.

A slope intercept calculator can be used to calculate the equation of the line according to the above steps.

• Point slope form

The other technique of calculating the equation of the straight line is the point-slope form. It is similar to the slope-intercept form but it deals with the slope of the line and points of the line while the slope-intercept form deals with the slope and y-intercept of the line.

The general equation of the point-slope form is:

y – y1 = m * (x – x1)

• m = the slope (inclination or steepness) of the line.
• x1 & y1 = the coordinate points of the line.
• x & y = the fixed points of the line.

The equation of the line can be calculated with the help of the point slope form equation easily by following the below steps.

• First of all, calculate the slope of the line by using the coordinate points.
• After that take a set of given points i.e., (x1, y1).
• Substitute the value of slope and (x1, y1) to the formula of the point slope form.
• The generated equation will be the linear equation of the line.

How to evaluate the equation of the line?

Here are a few examples of calculating the equation of the line.

Example 1: For slope intercept form

Calculate the linear equation of the line by using the slope intercept form (two points method). If the coordinate points of the line are:

(x1, y1) = (-12, 16) & (x2, y2) = (8, 14)

Solution

Step 1: Take the given information of the coordinate points of the line.

x1 = -12, x2 = 8, y1 = 16, y2 = 14

Step-2: First of all, evaluate the slope (steepness) of the line with the help of the coordinate points of the line.

The formula for calculating the slope of the line

Slope = m = [y2 – y1] / [x2 – x1]

Substitute the given values

Slope = m = [14 – 16] / [8 – (-12)]

Slope = m = [14 – 16] / [8 + 12]

Slope = m = [-2] / [20]

m = -2/20

m = -1/10 = -0.1

Step 3: Now calculate the y-intercept (y value) of the line by using the calculated slope and one pair of points to the general expression of the slope intercept form.

The general expression of the slope intercept form.

y = m * x + b

Substitute m = -0.1 and (x1, y1) = (-12, 16)

16 = -0.1 * (-12) + b

16 = 1.2 + b

16 – 1.2 = b

b = 14.8

Step 4: Now substitute the calculated values of the slope of the line and y-intercept of the line to the general expression of the slope intercept form to get the linear equation of the line.

y = m * x + b

y = -0.1 * x + 14.8

y = -0.1x + 14.8

Hence, the above expression is the linear equation of the line.

Example 2: For point slope form

Calculate the linear equation of the line by using the slope intercept form (two points method). If the coordinate points of the line are:

(x1, y1) = (-2, -26) & (x2, y2) = (12, 14)

Solution

Step 1: Take the given information of the coordinate points of the line.

x1 = -2, x2 = 12, y1 = -26, y2 = 14

Step-2: First of all, evaluate the slope (steepness) of the line with the help of the coordinate points of the line.

The formula for calculating the slope of the line

Slope = m = [y2 – y1] / [x2 – x1]

Substitute the given values

Slope = m = [14 – (-26)] / [12 – (-2)]

Slope = m = [14 + 26] / [12 + 2]

Slope = m = [40] / [14]

m = 40/14

m = 20/7 = 2.86

Step 3: Now write the general expression of the point slope form for calculating the equation of the line.

(y – y1) = m * (x – x1)

Step 4: Now put the calculated value of the slope of the line to the general expression of the point slope form and any pair of the coordinates points of the line to determine the straight line equation of the line.

The formula of the point slope form.

(y – y1) = m * (x – x1)

Substitute m = 2.86 and (x1, y1) = (-2, -26)

(y – (-26)) = 2.86 * (x – (-2))

(y + 26) = 2.86 * (x + 2)

(y + 26) = 2.86 * x + 2.86 * 2

y + 26 = 2.86x + 5.72

y + 26 – 2.86x – 5.72 = 0

y – 2.86x + 20.28 = 0

2.86x – y – 20.28 = 0

Wrap up

In this lesson, we have covered all the basic intent of calculating the equation of the line by using the slope intercept form and the point slope form. Now you can easily determine the linear equation of the line by following the above examples.