The following is a system of simultaneous linear equations to allocate costs using the reciprocal method. Matrix algebra is not required. The following costs were incurred in three operating departments and three service departments in Westmoreland Company.

$\backslash$table$[[$Department$,$Direct Costs,Label$],[$Subassemblies$,$$$550,000,$P$1],[$Final assembly,$775,000,$P$2],[$Marketing$,285,000,$P$3],[$Building occupancy,$85,000,$S$1],[$Research & development,$120,000,$S$2],[$Supervision$,45,000,$S$3]]$

Use of services by other departments is as follows.

$\backslash$table$[[,$User Departments$],[\backslash$table$[[$Service$],[$Cost Center$]],\backslash$table$[[$Sub$-],[$assemblies$]],\backslash$table$[[$Final$],[$Assembly$]],$Marketing,$\backslash$table$[[$Building$],[$Occupancy$]],$R&D$,$Supervision,$],[\backslash$table$[[$Building$],[$occupancy$]],0\mathrm{.}30,0\mathrm{.}25,0\mathrm{.}20,--,0\mathrm{.}15,0\mathrm{.}10,],[$R&D$,0\mathrm{.}50,0\mathrm{.}50,---,--,--,--,],[$Supervision$,0\mathrm{.}20,0\mathrm{.}30,0\mathrm{.}20,0\mathrm{.}10,0\mathrm{.}20,--,]]$

The equation for department S$2($research and development$)$ is: