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If an equation cannot be solved by Mathlab, it is possible that it is not in proper form. Any equation that is not in proper form can cause errors or miscalculations.
interpolation is basically following a trend. it is needed because one cannot always get enough data to get to where u want to go. GPS units "interpolate" when they loose satellite reception which is why they appear to be tracking in tunnels.
Interpolation in general is a way to determine intermediate values from a set of coordinates. Linear interpolation would be to fit a single linear function to the entire set of coordinates. Piecewise linear interpolation would then be to determine intermediate values from the set of coordinates by fitting linear functions between each set of coordinates. Connect-the-dots so to speak.
Linear interpolation is used as a method used in mathematics of constructing a curve that has the best fit to a series of points of data using linear polynomials.
Interpolation is filling in the data points between the data that has already been collected. Extrapolation is filling in data points beyond the data that has already been collected, or extending the data.