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 coordinatesx is the point to perform the interpolationy is the interpolated valueGreat learning in high school using simple cuesIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.Book a Free Trial ClassExamples 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 + 4y = 8Therefore, the value of y is 8.Example 2: Calculate the estimated height of the boy in the fourth position.position(x)x1 2 35Height in feet (y)y 3 4.556Solution: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 + 0.5y = 5.5Therefore, 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 = 5Therefore, the value of y is 5FAQs 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 coordinatesx is the point to perform the interpolationy is the interpolated valueWhat 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 + 2y = 7Therefore, the value of y is 7

What is Linear Interpolation? Linear interpolation is a method of estimating a value between two known data points. It assumes a straight line between these points and finds values along this line. This technique is widely used in various fields, including mathematics, engineering, and computer graphics. How to Calculate Linear Interpolation To perform linear interpolation, we use the equation of a straight line passing through two points. This allows us to estimate values between these points or even extend beyond them (extrapolation). Formula The formula for linear interpolation is: \[ y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{(x_2 - x_1)} \] Where: \((x_1, y_1)\) and \((x_2, y_2)\) are the known points \((x, y)\) is the point we're trying to find \(x\) is the known x-value for which we want to find the corresponding y-value Calculation Steps Identify the two known points \((x_1, y_1)\) and \((x_2, y_2)\) Determine the x-value for which you want to find the y-value Plug these values into the formula Solve for y Example and Visual Representation Let's interpolate between the points (2, 4) and (6, 10) to find y when x = 4. Using our formula: \[ y = 4 + \frac{(4 - 2)(10 - 4)}{(6 - 2)} = 4 + \frac{2 \cdot 6}{4} = 4 + 3 = 7 \] So, when x = 4, y = 7 Here's a visual representation of this linear interpolation: x y 0 4 8 12 2 4 6 8 (2,4) (6,10) (4,7) Calculation Steps: 1. Points: (2,4) and (6,10) 2. Slope = (10-4)/(6-2) = 1.5 3. y = y₁ + m(x-x₁) 4. y = 4 + 1.5(4-2) 5. y = 4 + 3 = 7 Result: (4,7) Rise = 3 Run = 2 In this graph, the blue dots represent our known points (2, 4) and (6,

You are given the X, Y Coordinates of 2 points, (99.5,39.5) and (201.1,45.7) and asked to find the Y coordinate of an intermediate point with an X coordinate of 115.7. How will you solve this problem?The answer is Interpolation!Interpolation is the method of estimating ‘unknown values’ that fall between ‘known values’.In this blog post, I will explain 2 Excel formulas to perform Linear interpolation to find the unknown.Table of ContentsLinear Interpolation FormulaInterpolation with FORECAST.LINEAR functionCalculating Y values against X values at regular intervalsMethod 1Method 2Linear Interpolation FormulaIf there are 2 known points denoted by the coordinates (x1, y1) and (x2 y2), the y coordinate of x in between x1 and x2 can be found out by the following formulaSee the following problem. If x1 = 99.5, y1 = 39.5, x2 = 201.1 and y2 = 45.7, what is the y coordinate of the point with x coordinate,115.7?When the above values are supplied into the Linear Interpolation formula, the formula to find the y coordinate will become,=(39.5*(201.1-115.7)+45.7*(115.7-99.5))/(201.1-99.5)In our Excel worksheet, x1 is in B3, y1 in C3, x2 in B6, y2 in C6 and x is in E3. So, the Excel formula equivalent to the above formula is,=(C3*(B6-E3)+C6*(E3-B3))/(B6-B3)and the formula evaluates to 40.49.Interpolation with FORECAST.LINEAR functionFORECAST.LINEAR function in Excel is used to predict a linear trend based on known values and the same can be used to perform Interpolation.Let’s solve the same problem that we discussed above, using FORECAST.LINEAR function.x1 = 99.5 (B3), y1 = 39.5 (C3), x2 = 201.1 (B4), y2 = 45.7 (C4) and we need to find the y when x = 115.7 (E3)FORECAST.LINEAR has 3 arguments x, known_ys, and known_xs.Here, x is 115.7, the value in the cell E3.known_ys are 39.5 and 45.7, the values in the cells C3 and C4.known_xs are 99.5 and 201.1, the values in the cells B3 and B4.So, the formula to calculate y value is,=FORECAST.LINEAR(E3,C3:C4,B3:B4)and the formula will return 40.49.Calculating Y values against X values at regular intervalsFollowing is another problem where we need to find the Y values against X values at 10-meter intervals (Column E).Method 1Solution using Linear Interpolation Formula.We