# Linear Interpolation Formula

Linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

Formula of Linear Interpolation

$$\begin{array}{l}y=y_{1}+\frac{\left(x-x_{1}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}\end{array}$$

Where,

$$\begin{array}{l}x_{1}\end{array}$$
and
$$\begin{array}{l}y_{1}\end{array}$$
are the first coordinates
$$\begin{array}{l}x_{2}\end{array}$$
and
$$\begin{array}{l}y_{2}\end{array}$$
are the second coordinates
x is the point to perform the interpolation
y is the interpolated value.

### Solved Examples

Question: Find the value of y at x = 4 given some set of values (2, 4), (6, 7)?

Solution:

Given the known values are,
x = 4 ; x1 = 2 ; x2 = 6 ; y1 = 4 ; y2 = 7

The interpolation formula is,

y =

$$\begin{array}{l}y_{1}\end{array}$$
+
$$\begin{array}{l}\frac{(x-x_{1})(y_{2}-y_{1})}{x_{2}-x_{1}}\end{array}$$

y = 4 +

$$\begin{array}{l}\frac{(4-2)(7-4)}{6-2}\end{array}$$

y = 4 +

$$\begin{array}{l}\frac{6}{4}\end{array}$$

y =

$$\begin{array}{l}\frac{11}{2}\end{array}$$