Linear Interpolation

Linear interpolation: how to approach it

Linear interpolation finds an intermediate value when the change is assumed to be constant between two known points. The key idea is that values lie on a straight line, so the fraction of the horizontal distance matches the fraction of the vertical change.

1) Identify the two known points

Write the data as [(x_1, y_1) \text{ and } (x_2, y_2).] Make sure you know which value you are trying to find: the output at a given input, or the input corresponding to a given output.

2) Compute the position fraction

Find how far the target is from the first point compared with the full interval: [ \frac{x - x_1}{x_2 - x_1}. ] This fraction tells you how much of the total change to take.

3) Apply the same fraction to the output change

Add that same proportion of the difference in y-values: [ y = y_1 + \frac{x - x_1}{x_2 - x_1}(y_2 - y_1). ] If the problem gives a target output and asks for the corresponding input, rearrange the same relationship to solve for (x).

4) Simplify carefully

Reduce fractions, combine integers, and leave the result in simplest exact form unless decimals are requested.

Check

Your answer should lie between the two known values when the target is between them. Also, plugging your result back into the proportional relationship should match the given data.