Apartments in complex

A developer offers for sale apartments in a newly built residential complex. In his offer there are the following three types of apartments:

MINI – 3 windows, has one balcony, has no garage parking space;
MEDIUM – 6 windows, has no balcony, has one garage parking space;
MAXI – 6 windows, has one balcony and two garage parking spaces.

When we walked around the building, we counted 51 windows, 9 balconies, and 10 garage parking spaces.

How many apartments of type MEDIUM are in the residential complex?

Final Answer:

b =  2

Step-by-step explanation:

$$\begin{gathered} 51=3\cdot a+6\cdot b+6\cdot c \\ 9=1\cdot a+0\cdot b+1\cdot c \\ 10=0\cdot a+1\cdot b+2\cdot c \\ 3a+6b+6c=51 \\ a+c=9 \\ b+2c=10 \\ Row2-\frac{1}{3}\cdot Row1\to Row2 \\ -2b-c=-8 \\ Row3-\frac{1}{-2}\cdot Row2\to Row3 \\ 1.5c=6 \\ c=\frac{6}{1.5}=4 \\ b=\frac{-8+c}{-2}=\frac{-8+4}{-2}=2 \\ a=\frac{51-6b-6c}{3}=\frac{51-6\cdot 2-6\cdot 4}{3}=5 \\ a=5 \\ b=2 \\ c=4 \end{gathered}$$

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