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. Let us learn more about the linear interpolation formula in this section.
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) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
Linear Interpolation Formula
The formula to calculate linear interpolation is:
Linear Interpolation(y) = \(y_{1}+( x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
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
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:
x = 6 ; \(x_1\) = 3 ; \(x_2\) = 6 ; \(y_1\) = 4 ; \(y_2\) = 8 (given)
Using linear interpolation formula,
Linear Interpolation(y) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
Put the values,
\(y= 4+(6-3)\dfrac{8-4}{6-3}\)
y = 4 + 3(4/3)
y = 4 + 4
y = 8
Therefore, 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 feet (y) | y | 3 |
4.5 | 5 | 6 |
Solution:
x = 4 ; \(x_1\) = 3 ; \(x_2\) = 5 ; \(y_1\) = 5 ;\(y_2\) = 6 (given)
Using linear interpolation formula,
Linear Interpolation(y) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
Put the values,
\(y= 5 + (4 - 3)\dfrac{(6 - 5)}{(5-3)}\)
y = 5 + 1(1/2)
y= 5 + 0.5
y = 5.5
Therefore, the height of the boy in the fourth position is 5.5 feet.
Example 3: Find the value of y if x = 8 and some set of values are given as (5, 3.5), (10, 6)?
Solution:
x = 8 ; \(x_1\) = 5 ; \(x_2\) = 10 ; \(y_1\) = 3.5 ; \(y_2\) = 6 (given)
Using linear interpolation formula,
Linear Interpolation(y) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
Put the values,
\(y= 3.5 + (8- 5)\dfrac{(6 - 3.5)}{(10- 5)}\)
y = 3.5 +3(2.5/5)
y = 3.5 + 3(1/2)
y = 3.5 + 1.5
y = 5
Therefore, the value of y is 5
FAQs on Linear Interpolation Formula
What is Meant by Linear Interpolation Formula?
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. The linear interpolation formula is used for data forecasting, data prediction, mathematical and scientific applications and, market research, etc. The formula is (y) = \(y_{1}+\frac{\left(x-x_{1}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}\)
What is the Formula to Calculate Linear Interpolation Formula?
The formula to calculate linear interpolation is:
Linear Interpolation(y) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
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
What are the Uses of Linear Interpolation Formula?
The linear interpolation formula is helpful in determining the values between any two given points. Hence, linear interpolation is also considered as a method of filling in the gaps for any value in a table format. The formula helps in creating a straight line along with the given points on both the negative and positive sides.
Using the Linear Interpolation Formula, Find the Value of y when x = 8 along with Coordinates (7,5) and (10,9)
x = 8 ; \(x_1\) = 6 ; \(x_2\) = 10 ; \(y_1\) = 5 ; \(y_2\) = 9 (given)
Using linear interpolation formula,
Linear Interpolation(y) = \(y_{1}+ (x-x_{1})\dfrac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)
Put the values,
\(y= 5+(8-6) \dfrac{9-5}{10-6}\)
y = 5+ 2(4/4)
y = 5 + 2
y = 7
Therefore, the value of y is 7
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