Equation of the Line: Point Slope and Slope-Intercept Form

Published by Kabir Chibber on March 22, 2022March 22, 2022

In mathematics and physics, the equation of the line is used frequently. One-degree variables terms are referred to as an equation. The equation of the line can be determined by using the slope and points of the line. 

There are various methods to evaluate the equation of the line. In this post, we will discuss two methods to determine the equation of the line. The point-slope form and slope-intercept form are methods to evaluate the equation of the line. 

What is the point–slope form and slope–intercept form?

There are various methods to calculate the equation of the line. The point-slope form and slope-intercept form are methods to find the equation of the line by using points and slope of the line. Both methods are essential for the calculation of the equation of the line.

Point Slope Form for Equation of the Line

The point-slope form is a way to determine the equation of the line by using the line’s slope and points. The point-slope form evaluates the equation of the line by using a formula. First of all, you have to determine the slope of the line by using the given points of the line.

The equation used to determine the slope of the line is given below.

m = y – y₁ / x – x₁

Using the slope equation, we can distinguish the point-slope form. 

y – y₁ = m (x – x₁)• m is the slope of the line.• x₁ and y₁ are the points of the line.• x and y are the coordinates points of the line.

Slope Intercept Form for Equation of the Line

The slope-intercept form is another method to determine the equation of the line. In this method, the slope and y-intercept are used to evaluate the equation of the line. First of all, you have to calculate the slope of the line and the y-intercept. 

Then use the formula of slope-intercept form and place the calculated slope and y-intercept to get the equation of the line. The equation of slope-intercept form that is used to evaluate the equation of the line is given below.

y = mx + c• m is the slope of the line.• x and y are the fixed points.• c is the y-intercept of the line.

You can use a slope intercept form calculator to determine the equation of the line in a few seconds. This calculator evaluates the equation of the line according to the equation of the slope-intercept form. Follow the below steps to use this calculator.

Step 1: Select the option, i.e., for two points, slope and one point, or y-intercept and slope.

Step 2: Enter the points in case you select two points option otherwise enter slope along with point or y-intercept.

Step 3: Hit the calculate button below the input boxes.

You will get the step-by-step solution of the given problem in a couple of seconds along with the graph.

Evaluate Equation of the Line using Point-Slope and Slope-Intercept form

Following are a few examples solved by using the point-slope form and the slope-intercept form to evaluate the equation of the line.

Example 1: By using point-slope form

Calculate the line’s equation of (3, -21) and (9, 23) by using a point-slope form.

Solution

Step 1: Write the given points of the line.

x₁ = 3, x₂ = 9, y₁ = -21, y₂ = 23

Step 2: First of all, evaluate the slope of the line by using the given points.

Slope = m = (y₂ – y₁) / (x₂ – x₁)

Slope = m = (23 – (-21)) / (9 – 3)

Slope = m = (23 + 21) / (6)

Slope = m = 44/6

Slope = m = 22/3

Slope = m = 7.33

Step 3: Now write the point-slope form’s equation.

(y – y1) = m (x – x1)

Step 4: Now put the calculated slope in the above equation and use any pair of numbers.

(y – y1) = m (x – x1)

(y – (-21)) = 7.33(x – 3)

y + 21 = 7.33x – 21.99

y + 21 – 7.33x + 21.99 = 0

y – 7.33x + 42.99 = 0

7.33x – y – 42.99 = 0

7.33x – 42.99 – y = 0

You can also use a point slope form calculator to verify the result.

Example 2

Calculate the line’s equation of (53, -31) and (109, 73) by using a point-slope form.

Solution

Step 1: Write the given points of the line.

x1 = 53, x2 = 109, y1 = -11, y2 = 73

Step 2: First of all, evaluate the slope of the line by using the given points.

Slope = m = (y2 – y1) / (x2 – x1)

Slope = m = (73 – (-31)) / (109 – 53)

Slope = m = (73 + 31) / (56)

Slope = m = 104/56

Slope = m = 26/14 = 13/7

Slope = m = 1.8571

Step 3: Now write the point-slope form’s equation.

(y – y1) = m (x – x1)

Step 4: Now put the calculated slope in the above equation and use any pair of numbers.

(y – y1) = m (x – x1)

(y – (-31)) = 1.8571(x – 53)

(y + 31) = 1.8571x – 98.4263

y + 31 – 1.8571x + 98.4263 = 0

y – 1.8571 x + 129.4263 = 0

1.8571x – y – 129.4263 = 0

1.8571x – 129.4263 – y = 0

Example 3: By using Slope-intercept form

Calculate the line’s equation of (-5, -1) and (10, 3) by using a slope-intercept form.

Solution

Step 1: Write the given points of the line.

x1 = -5, x2 = 10, y1 = -1, y2 = 3

Step 2: First of all, evaluate the slope of the line by using the given points.

Slope = m = (y2 – y1) / (x2 – x1)

Slope = m = (3 – (-1)) / (10 – (-5))

Slope = m = (3 + 1) / (10 + 5)

Slope = m = 4/15

Slope = m = 0.2667

Step 3: Now write the slope-intercept form’s equation.

y = mx + b

Step 4: Now calculate the y-intercept by using any pair of numbers.

y = mx + b

3 = 0.2667(10) + b

3 = 2.667 + b 

3 – 2.667 = b

0.333 = b

b = 0.333

Step 5: Now put the values in the slope-intercept form’s equation.

y = mx + b

y = 0.2667x + 0.333

Now you know the two methods to determine the equation of the line. You can use the calculators to solve them too. You can also learn about the divisibility rules that help in factorisation.