I just saw this question and I will come back to elaborate.  The use of double reciprocal plots is problematic and can mislead an investigator to the wrong conclusion. (Traylor, Stynes J. Am. Chem. Soc circa 1985 or 1986).  Lineweaver-Burke plots are out  dated as there are a myriad of analysis software for non-linear fits of the experimental data. 

I agree with Gert Van der Zwan's answer, as well as his comment that you are very vague about what kind of spectrophotometric data you have collected.  Ideally, you will have collected a set of time ooints A652(t) at incremental times t (t1, t2, t3, ..., or a continuous curve of A(t)) vs. t).  If this curve looks like a straight line (the product is highly colored so you get substantial increases in A652 for small changes in substrate concentration, [S]), then the slope of that line is the velocity dA/dt.  If the time plot is curved, you need to get the initial slope for early values of t:  delta A/delta t ~ dA/dt for t values approaching t=0.  If, as in the old days, you have only two points, A0 at t=0 and A(t1) at t=t1, then you have to assume you are in the initial rate region and use  dA/dt ~ (A(t=0) - A(t=t1))/(0 - t1).  In any case, you can get Km without converting dA/dt to dv/dt, where v = d{S]/dt using Beers' Law:  [P] = A652/e652 (where e652 is the absorptivity of the product P.  If one mole of substrate S turns into one mole of product P, then v = -d[S]/dt = d[P]/dt.  You would have to verify that the change in [S] is small, that  is, [S(t1)] = {S0] - [P[t1] is not more than a few percent from [S0].  You then either get a non-linear fit to v = Vmax[S0]/(Km + [S0}), or you use one of the three ways you can get a straight line from the Michaelis-Menten equation and fit y = a + bx for the relevant forms (although if you get a good fit to the MM eqn I don't see why you need to use a double reciprocal plot except for demonstration purposes).

Gert Van der Zwan,Theo Nikiforov,David  William Thompson and Arnold Wishnia thanks alot for your help. I have collected following data

I am about to try few more changes to my experimental conditions along trying to focus on the insights provided by your esteemed suggestions. Will get back to your help if i need more comprehension on the matter. Thanks again. 

I'm sorry I  forgot to repeat what others have said, that you have to get a separate time curve for each of a set of initial concentrations [S0]1, [S0]2, ...  spanning the range 0.1 to 10 Km (or at least 0.2 to 3 Km).  The problem with fittings to any of the linearized forms is that you have to give the proper weights to the separate points, whereas equal (i. e., unit) weighting is ok for a fitting to the non-linear form of the MM eqn. 

It is possible to get Vmax and Km from a fitting of a single time curve (if you're lucky) to the integrated  form of the MM eqn, but that fitting is a lot more complicated than the fitting to the standard MM eqn (and I'm sorry if that last comment only confuses the issue).

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I don't have anything to add to the excellent points made by everyone, but I will try to describe the procedure in a simple way, so that it is easy to follow for anyone who does not have any experience with such experiments.

1. Carry out enzyme reaction in which substrate concentration is, say, S1, and enzyme concentration is much smaller than S1.

2. Follow this reaction using a spectrophotometer by measuring the absorbance (A365 or whatever is appropriate) continuously over time.

3. Calculate the slope of this continuous curve in the initial region. Make sure that the region is linear or a straight line. This slope is now V1.

4. Repeat steps 1 - 3 using different concentrations of substrate - S2, S3, S4 etc, and obtain the slopes - V2, V3, V3 etc., for each.

5. Use the substrate concentrations (S1, S2 ..) as the x values and the slopes (or 'Velocities', V1, V2 ..) as the y values in Prism software and apply the Michealis Menten function to fit the data. 

6. The results or output of this fitting will be the Km and Vmax values.

I am not sure if Prism has a built-in way to make the double reciprocal plots. To make these:

1. Calculate the reciprocal of substrate concentrations - 1/S1, 1/S2 etc.

2. Calculate the reciprocal of the slopes or velocities - 1/V1, 1/V2 etc.

3. Plot the reciprocal of substrate concentrations (as x) vs the reciprocal of slopes (as y). 

4. Fit a linear regression (y = mx + c) for this data.

5. The value of c will give you "-1/Km" and the value of m will give you "Km/Vmax"

As others have said, using the double reciprocal plot to calculate these values is not reliable, and was only used when there were no computers for direct calculation of Michealis Menten curve which requires 'non-linear' regression analysis - which can now be done using software such as Prism.

I hope this helps.

Using the Advanced Statistical Programming Language R in Rstudio. 

First Install R, and Install Rstudio. 

You can actually compute Vmax and Km directly from R, and make the plot with the best fitting Michaelis-Menten curve using Rstudio, and then you can add the curve to the scatterplot.

First make an Excel file with two experimental columns: "S" for substrate Concentration, and "v" for velocity. Save your file as a comma separated values (.csv) file, and then load it into R. I have provided you with the file. 

I have some data from nls tools in R. The dataset is vmkm.

Load the file vmkm.csv into R like so.

Execute these commands in Rstudio:

> mm View(mm)

> plot(v ~ S, pch = 16, data = mm, main = "Michaelis Menten Plot", xlab = "Substrate Concentration", ylab = "Velocity")

> model summary(model) 

# This computes nonlinear regression of the michaelis menten equation:

> curve(1.5460*x/(2.5861 + x), col = "red", add = TRUE)

# This is the equation of the curve. Notice 1.5460 = Vmax, and 2.5861 = Km which is what I retrieved, so I am adding the Michealis Menten curve to the scatter-plot. 

# The curve function just assumes that "x" is always your horizontal axis. It is substrate concentration, so you see why I used x instead of [S]. 

R can also compute the LineWeaver Burk plot for you. 

> mm$reciprocal_S  mm$reciprocal_v mm 

> plot(mm$reciprocal_v ~ mm$reciprocal_S, pch = 16, main = "Experimental Line Weaver Burk Plot", xlab = "1 / [S]", ylab = "1 / [v]")

> abline(lm(mm$reciprocal_v ~ mm$reciprocal_S))

# This plots the line.

> summary(lm(mm$reciprocal_v ~ mm$reciprocal_S))

# This gives you equation of the line by linear regression.

y = 1.6286(x) + 0.6284 

If you do some algebra, you figure out that according to the LineWeaver Burk Plot Km = 2.3866, and Vmax = 1.4654

The nonlinear curve fitting didn't exactly give those values. But from the summary of the model if you check the standard errors, they are within the standard errors, so you could do the curve function to the scatterplot and fit the more accurate Michealis Menten curve now that you know Vmax and Km from the lineweaver burk plot. 

Nevermind, it appears that the non-linear regression model is slightly better fit to the scatterplot. 

I figured out this solution by consulting these R instructional documents: 

https://ww2.coastal.edu/kingw/statistics/R-tutorials/simplenonlinear.html

https://rpubs.com/RomanL/6752

Yes, I know the substrate concentration should go to 10. Here it goes to 2 in the dataset that I selected, but you can copy and paste these commands to compute Vmax and Km to any dataset. 

Use a nonlinear regression using the M-M equation as a model. You can add the zero (zero substrate & zero Vo) and try a suitable software such as Origin (it has a hyperbolic fit), QtiPlot, SciDavis or GnuPlot. For the last three options you'll have to write the equation. As mentioned above, try to choose [S] below (0.1 or 0.5 Km and up to 10 Km is that is possible. If you don't have an idea about the Km value perform some enzyme assays and a direct linear plot (Article The direct linear plot. A new graphical procedure for estima...

Dear Adron Ung ,

Can you tell me how you calculate velocity from absorbance?