Hi, i'm stuck on a part of this question, we have to solve using gauss jordan elimination: a corporation wants to lease a fleet of 12 airplanes with a combined carrying capacity of 220 passengers. the three available types of planes carry 10,15,20 passengers. how many of each type of plane should be leased?

Do common sense has anything to do with this question? Because 
corporation might forget the arbitrary 12 airplane rule 
and have 11 times 20 = 220. 
But perhaps they get a discount if all 12 hangars are used! 

Playing by the rules of the question:- 
Let a, b, c be the aeroplanes that carry 
10, 15, and 20 passengers, respectively, 

a + b + c = 12 
10a +15b +20c = 220 

We may rearrange these as 
4a + 4b + 4c = 48 
2a + 3b + 4c = 44 
Now subtract 
2a + b = 4 

There appear to be 3 options 
a = 0, b = 4 which means 220 = 0*10 + 4*15 + 8*20 
a = 1, b = 2 which means 220 = 1*10 + 2*15 + 9*20 
a = 2, b = 0 which means 220 = 2*10 + 0*15 + 7*20 

If you dismiss the ones with zeros it may be that 
the expected answer to "How many of each plane" is 

One that carries 10, 2 that carries 15, and 9 that carries 20