Estimating misclassification rate from summary classification values

I'm trying to work out the misclassification rate in a published study, where the results do not include this value. However, I have the sensitivity, specificity, positive predictive value and negative predictive value results.

The results are:

Sensitivity     Specificity     PPV      NPV
69.9            99.3            95.6     93.7

The sample size is 860. I figure this can be solved using a set of 4 simultaneous equations, but my algebra isn't up to the task.

If we use the following letters for replacement in the equations:

• a = true positives
• b = false positives
• c = false negatives
• d = true negatives

Then the set of simultaneous equations appears to be:

a/(a+c)=.699 (i.e. sensitivity)
b/(b+d)=.993 (i.e. specificity)
a/(a+b)=.956 (i.e. PPV)
d/(c+d)=.937 (i.e. NPV)

The fractional terms in the equations have been driving me nuts. Surely someone else must have had the same problem in the past and solved it, but I have been unable to find a ready reckoner on how to solve this problem. Once I know what the four values are, I can easily calculate the misclassification rate.

I would appreciate any assistance on this back calculation problem. I've posted here as I'm assuming an epidemiologist or biostatistician is the most likely type of person to answer and they may not read the mathematics questions.