Linear Interpolation Formula

Linear Interpolation Formula is a method that constructs the new data points from the given set of data points. Linear interpolation is used for fitting curves using linear polynomials. It finds the unknown values in the table.

What is the Linear Interpolation Formula?

The formula of linear interpolation is given by

Linear Interpolation (y) = y₁ + [(x-x₁) × (y₂-y₁)]/ (x₂-x₁)

Where,

(x1,y1) & (x2,y2) are coordinates. 

x is the point to perform interpolation.

y is the interpolated value.

Applications of Linear Interpolation Formula

1. Data Prediction.
2. Data Forecasting.
3. Mathematical applications in the marketing field.
4. Scientific applications in the research field.

Sample Questions on Linear Interpolation Formula

To get more understanding about the linear interpolation method let's look into a few examples.

Question 1: Find the value of y if x=5 and some set of points are (1,4) and (6,9)

Solution:

Given,

(x₁,y₁) = (1,4)

(x_2,y₂) = (6,9)

x = 5

From Linear Interpolation formula,

y = y₁+ (((x-x₁) x (y₂-y₁))/ (x₂-x₁))

y = 4+(((5-1) x (9-4))/(6-1))

   = 4+((4x5)/5)

   = 4+(20/5)

   = 4+4

   = 8

So new data point (x, y) is (5,8)

Question 2: Find the new data point (x, y) in which x=2 from the data points (1,3) and (4,12)

Solution:

Given,

(x₁,y₁) = (1,3)

(x₂,y₂) = (4,12)

x = 2

From Linear Interpolation formula,

y = y₁+ (((x-x₁) x (y₂-y₁))/ (x₂-x₁))

y = 4+(((2-1) x (12-3))/(4-1))

   = 4+((1x9)/3)

   = 4+(9/3)

   = 4+3

   = 7

So new data point (x, y) is (2,7)

Question 3: Find the value of y if x=4 from the set of points are (3,4),(5,9)

Solution:

Given,

(x₁,y₁) = (3,4)

(x₂,y₂) = (5,9)

x = 4

From Linear Interpolation formula,

y = y₁+ (((x-x₁) x (y₂-y₁))/ (x₂-x₁))

y = 4+(((4-3) x (9-4))/(5-3))

  = 4+((1x5)/2)

  = 4+(5/2)

  = 4+2.5

  = 6.5

So new data point (x, y) is (4,6.5)

Question 4: Find the new data point (x, y) in which x=6 from the data points (5,10) and (10,10)

Solution:

Given,

(x₁,y₁) = (5,10)

(x₂,y₂) = (10,10)

x = 6

From Linear Interpolation formula,

y = y₁+ (((x-x₁) x (y₂-y₁))/ (x₂-x₁))

y = 10+(((6-5) x (10-10))/(10-5))

   = 10+((1x0)/5)

   = 10+(0/5)

   = 10+0

   = 10

So new data point (x, y) is (6,10)

Question 5: Find the value of y if x=2.5 from the set of points are (0,5.5) and (5,7.5)

Solution:

Given,

(x₁,y₁) = (0,5.5)

(x₂,y₂) = (5,7.5)

x = 2.5

From Linear Interpolation formula,

y = y₁+ (((x-x₁) x (y₂-y₁))/ (x₂-x₁))

y = 5.5+(((2.5-0) x (7.5-5.5))/(5-0))

   = 5.5+((2.5x2)/5)

   = 5.5+(5/5)

   = 5.5+1

  = 6.5

So new data point (x, y) is (2.5,6.5)