Chamber

In the chamber, light is broken, and all from it must be taken at random. Socks have four different colors. If you want to be sure of pulling at least two white socks, we have to bring them out 28 from the chamber. To have such certainty for the pair of gray socks, we must bring them also 28, for just a pair of black socks 26 and a pair of blue socks 34. How much are the full socks in the chamber?

Correct answer:

x =  36

Step-by-step explanation:

$$\begin{gathered} x=a+b+c+d \\ a+b+c=28-2 \\ b+c+d=28-2 \\ c+d+a=26-2 \\ d+a+b=34-2 \\ a+b+c+d-x=0 \\ a+b+c=26 \\ b+c+d=26 \\ a+c+d=24 \\ a+b+d=32 \\ Row2-Row1\to Row2 \\ a+b+c+d-x=0 \\ -d+x=26 \\ b+c+d=26 \\ a+c+d=24 \\ a+b+d=32 \\ Row4-Row1\to Row4 \\ a+b+c+d-x=0 \\ -d+x=26 \\ b+c+d=26 \\ -b+x=24 \\ a+b+d=32 \\ Row5-Row1\to Row5 \\ a+b+c+d-x=0 \\ -d+x=26 \\ b+c+d=26 \\ -b+x=24 \\ -c+x=32 \\ Pivot:Row2\leftrightarrow Row3 \\ a+b+c+d-x=0 \\ b+c+d=26 \\ -d+x=26 \\ -b+x=24 \\ -c+x=32 \\ Row4+Row2\to Row4 \\ a+b+c+d-x=0 \\ b+c+d=26 \\ -d+x=26 \\ c+d+x=50 \\ -c+x=32 \\ Pivot:Row3\leftrightarrow Row4 \\ a+b+c+d-x=0 \\ b+c+d=26 \\ c+d+x=50 \\ -d+x=26 \\ -c+x=32 \\ Row5+Row3\to Row5 \\ a+b+c+d-x=0 \\ b+c+d=26 \\ c+d+x=50 \\ -d+x=26 \\ d+2x=82 \\ Row5+Row4\to Row5 \\ a+b+c+d-x=0 \\ b+c+d=26 \\ c+d+x=50 \\ -d+x=26 \\ 3x=108 \\ x=\frac{108}{3}=36 \\ d=\frac{26-x}{-1}=\frac{26-36}{-1}=10 \\ c=50-d-x=50-10-36=4 \\ b=26-c-d=26-4-10=12 \\ a=0-b-c-d+x=0-12-4-10+36=10 \\ a=10 \\ b=12 \\ c=4 \\ d=10 \\ x=36 \end{gathered}$$

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