Favourite singers

A poll carried out among 200 students of one school found out which of the singers V. Neckář,
K. Gott, W. Matuška is the most popular. It brought the following results: Gott is popular with 78
students, Matuška with 75 students, and Neckář with 101. It was further found that all three of them are
popular with 28 students. Those who admire exactly two of these three singers are 22, of them exactly
half admire the pair Neckář and Matuška. Students who admire only Matuška are
7 fewer than those who admire only Gott.

(a) How many students admire V. Neckář?
(b) How many students admire K. Gott?
(c) How many students admire W. Matuška?
(d) How many students do not admire any of the mentioned?

Final Answer:

n =  101
g =  78
m =  75
x =  24

Step-by-step explanation:

$$\begin{gathered} s=200 \\ c=22/2=11 \\ 78=a+b+28+g_1 \\ 101=b+28+c+n_1 \\ 75=a+28+c+m_1 \\ {m}_1=g_1-7 \\ c=a+b \\ 78=a+b+28+g_1 \\ 101=b+28+11+n_1 \\ 75=a+28+11+m_1 \\ {m}_1=g_1-7 \\ 11=a+b \\ a+b+g_1=50 \\ b+n_1=62 \\ a+m_1=36 \\ {g}_1-m_1=7 \\ a+b=11 \\ Row3-Row1\to Row3 \\ -b-g_1+m_1=-14 \\ Row5-Row1\to Row5 \\ -g_1=-39 \\ Row3+Row2\to Row3 \\ -g_1+m_1+n_1=48 \\ Row4+Row3\to Row4 \\ {n}_1=55 \\ Row5-Row3\to Row5 \\ -m_1-n_1=-87 \\ Pivot:Row4\leftrightarrow Row5 \\ {m}_1=\frac{-87+n_1}{-1}=\frac{-87+55}{-1}=32 \\ {g}_1=\frac{48-m_1-n_1}{-1}=\frac{48-32-55}{-1}=39 \\ b=62-n_1=62-55=7 \\ a=50-b-g_1=50-7-39=4 \\ a=4 \\ b=7 \\ {g}_1=39 \\ {m}_1=32 \\ n=n_1+b+28+11=55+7+28+11=101 \end{gathered}$$

$g=g_1+a+b+28=39+4+7+28=78$

$m=m_1+a+28+11=32+4+28+11=75$

$x=s-(g_1+a+b+c+28+n_1+m_1)=200-(39+4+7+11+28+55+32)=24$

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