Linear Interpolation Formula
The linear interpolation formula is the simplest method that is used for estimating the value of a function between any two known values. Also, the linear interpolation formula is a method that is useful for curve fitting using linear polynomials. Basically, the interpolation method is used for finding new values for any function using the set of values. The unknown values in the table are found using the linear interpolation formula.
What is Linear Interpolation Formula?
The linear interpolation formula is used for data forecasting, data prediction, mathematical and scientific applications and, market research, etc. The linear interpolation formula can be used for finding the unknown values in the table. The formula for linear interpolation formula is given by:
Linear Interpolation(y) = \(y_{1}+\frac{\left(xx_{1}\right)\left(y_{2}y_{1}\right)}{x_{2}x_{1}}\)
where,
 \(x_1\) and \(y_1\) are the first coordinates
 \(x_2\) and \(y_2\) are the second coordinates
 x is the point to perform the interpolation
 y is the interpolated value
Let's take a quick look at a couple of examples to understand the linear interpolation formula, better.
Solved Examples Using Linear Interpolation Formula

Example 1: Find the value of y if x = 6 and some set of values are given as (3, 4), (6, 8)?
Solution:
To find: The value of y
x = 6 ; \(x_1\) = 3 ; \(x_2\) = 6 ; \(y_1\) = 4 ; \(y_2\) = 8 (given)
Using linear interpolation formula,
\(\text{Linear Interpolation Formula (y)}=y_{1}+\frac{\left(xx_{1}\right)\left(y_{2}y_{1}\right)}{x_{2}x_{1}}\)
Put the values,
\(y= 4+\frac{\left(6  3\right)\left(8  4\right)}{6  3}\)
\(y= 4+\frac{\left(3\right)\left(4\right)}{3}\)
\(y= 4+\frac{12}{3}\)
y = 4 + 4y = 8
Answer: The value of y is 8.

Example 2: Calculate the estimated height of the boy in the fourth position.
position(x) x 1 2 3 5 Height in feets(y) y
3
4.5 5 6 Solution:
To find: Height of the boy in the fourth position.
x = 4 ; \(x_1\) = 3 ; \(x_2\) = 5 ; \(y_1\) = 5 ;\(y_2\) = 6 (given)
Using linear interpolation formula,
\(\text{Linear Interpolation Formula (y)}=y_{1}+\frac{\left(xx_{1}\right)\left(y_{2}y_{1}\right)}{x_{2}x_{1}}\)
Put the values,
\(y= 5 + \frac{\left(4  3\right)\left(6  5\right)}{5  3}\)
\(y= 5+\frac{\left(1\right)\left(1\right)}{2}\)
y= 5 + 0.5
y = 5.5Answer: The height of the boy in the fourth position is 5.5 feet.