Question 843673
Using A, B, and C as the rates in gallons per hour for each of the pump named the same, the rates of any pumps operating at the same time are simply additive.  This allows three equations for the three combinations given in the description.


System:
A+B+C=3400
A+B=1900
A+C=2300


My method was to set up the matrix and to use just enough row operations to be able to back-substitute; although you could try using substitutions and resolving resulting equations in the old style if that were all you knew.


Matrix corresponding to the system listed above is
(   1,   1,   1,   3400 )
(   1,   1,   0,   1900 )
(   1,   0,   1,   2300 )

I omit the steps here, but here then are the results from the steps:
ANSWER:
{{{A=5200}}}
{{{B=1100}}}
{{{C=-2900}}}
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-
Notice that C is NEGATIVE, which means that pump C does NOT fill the tank, but 
REMOVES liquid from the tank.