[Example 5.5.4] $A, B, C, D, E$ 5 people participate in an exam, which has 7 questions, all of which are true/false questions. The scoring rule is: for each question, a correct answer earns 1 point, a wrong answer deducts 1 point, and no answer neither earns nor deducts points. The table below records the answers of $A, B, C, D, E$. It is known that $A, B, C, D$ each scored 2 points. How many points should $E$ get? What is the correct answer for each question?
\begin{tabular}{|c|c|c|c|c|c|}
\hline \begin{tabular}{c} 
Person \\
Question Number
\end{tabular} & $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline 1 & $\checkmark$ & $\checkmark$ & & $X$ & $\checkmark$ \\
\hline 2 & & $X$ & $\checkmark$ & $X$ & $\checkmark$ \\
\hline 3 & $X$ & $\checkmark$ & $X$ & $X$ & $X$ \\
\hline 4 & $\checkmark$ & $\checkmark$ & $X$ & $\checkmark$ & \\
\hline 5 & $X$ & $X$ & $\checkmark$ & $\checkmark$ & $\checkmark$ \\
\hline 6 & $\checkmark$ & $X$ & $X$ & & $\times$ \\
\hline 7 & $\checkmark$ & & $\checkmark$ & $\checkmark$ & $\checkmark$ \\
\hline
\end{tabular}
This problem is difficult to solve directly because we don't know the standard answers to each question. Taking question 5 as an example, both $A$ and $B$ think this question is incorrect, while $C$, $D$, and $E$ think it is correct. So, should this question be filled with “$X$” or “$\checkmark$”? This is not clear from the problem statement. The only way to determine the correct answer for each question is by analyzing the final scores and answer sheets of $A$, $B$, $C$, and $D$. Only by figuring out the standard answers to each question can we determine $E$'s score.

To achieve this, we assign a value to each question. If the conclusion of the $k$-th question is correct, we set $x_{k}=1$; if it is incorrect, we set $x_{k}=-1$.

According to this assignment, when the conclusion of the $k$-th question is correct, i.e., $x_{k}=1$, if someone judges it as correct, they get $x_{k}$ points; if they judge it as incorrect, they get $-x_{k}$ points. When the conclusion of the $k$-th question is incorrect, i.e., $x_{k}=-1$, if someone judges it as correct, they get $x_{k}$ points; if they judge it as incorrect, they get $-x_{k}$ points. We list the answer statuses of $A$, $B$, $C$, and $D$ in equations, and we know that these 4 people each scored 2 points. This gives us the following system of equations:
$$
\left\{\begin{array}{l}
x_{1}+0 x_{2}-x_{3}+x_{4}-x_{5}+x_{6}+x_{7}=2, \\
x_{1}-x_{2}+x_{3}+x_{4}-x_{5}-x_{6}+0 x_{7}=2, \\
0 x_{1}+x_{2}-x_{3}-x_{4}+x_{5}-x_{6}+x_{7}=2, \\
-x_{1}-x_{2}-x_{3}+x_{4}+x_{5}+0 x_{6}-x_{7}=2 .
\end{array}\right.
$$
Adding (1) + (2) + (3) + (4), we get
$$
x_{1}-x_{2}-2 x_{3}+2 x_{4}+0 x_{5}-x_{6}+x_{7}=8 .
$$

Since $x_{k}= \pm 1(k=1,2, \cdots, 7)$, and
$$
\left|x_{1}\right|+\left|x_{2}\right|+2\left|x_{3}\right|+2\left|x_{4}\right|+0\left|x_{5}\right|+\left|x_{6}\right|+\left|x_{7}\right|=8 \text{. }
$$

Comparing (5) and (6), and considering that $x_{k}= \pm 1$, we have
$$
x_{1}=1, x_{2}=-1, x_{3}=-1, x_{4}=1, x_{6}=-1, x_{7}=1 .
$$

From (5), we can determine the values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{6}, x_{7}$. Substituting these values into equation (1) gives $1+0+1+1-x_{5}-1+1=2$,
thus
$$
x_{5}=1 \text{. }
$$

From the above, we know that among the 7 judgment questions, the conclusions of questions 1, 4, 5, and 7 are correct, while the conclusions of questions 2, 3, and 6 are incorrect.

At this point, $E$ answered questions 1, 3, 5, 6, and 7 correctly, and answered question 2 incorrectly, so $E$ scored 4 points.