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Reciprocal Method of Cost Accounting with a Matrix Algebra equation

Warren Ltd. has two production departments, Building A and Building B, and two service departments, Maintenance
and Cafeteria. Direct costs for each department and the proportion of service costs used by the various departments
for the month of June follow:
Proportion of Services Used by

Department Direct Costs Maintenance Cafeteria Building A Building B
  Building A $ 505,000
  Building B 315,000
  Maintenance 217,000 â?" 0.2 0.6 0.2
  Cafeteria 165,000 0.7 â?" 0.1 0.2

Use the reciprocal method to allocate the service costs. (Matrix algebra is not required.) (Note: Due to rounding, the
cost allocations to the various departments may not add up to the total service department costs being allocated.)

Cost Allocated To:
From: Maintenance Cafeteria Building A Building B
Service Dept 217,000 165,000 0 0

Set up the equations:
Total service = Direct costs of the + Cost allocated
department costs service department to the service
   S1 (Maintenance) = $217,000 + 0.70 S2
   S2 (Cafeteria) = 165,000 + 0.20 S1

Substituting, the first equation into the second yields,

S2 = $165,000 + 0.20 ($217,000 + 0.70 S2)
S2 = $165,000 + $43,400 + 0.14 S2
.86S2 = $208,400
S2 = $242,326

I do not understand how S2 has been determined to be .86, can someone please explain it to me?

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Solution Summary

The following posting helps with a problem involving the reciprocal method for cost allocations.