在一次综合知识竞赛的决赛中，一个参赛队需要回答 25 道题。每答对一题得 5 分，答错一题扣 3 分。如果是部分正确，该队得 2 分。
在回答完 24 道题后，结果如下表所示。 | 题目数量 | | |
| :---: | :---: | :---: | :---: |
| 正确 | 部分正确 | 错误 | 得分 |
| $a$ | $b$ | $c$ | 79 | 如果该队在最后一题中回答错误，那么答对和部分正确的总题数将是答错题数的四倍。通过建立线性方程组，求 $a$、$b$ 和 $c$ 的值。

1. Form the first two linear equations from the table given for the first 24 questions.

Since the table shows the results after 24 questions, the total number of questions answered is 24:
$a + b + c = 24$

Given the scoring system (5 points for correct, 2 points for partially correct, and a 3-point deduction for wrong answers), the total score is 79:
$5a + 2b - 3c = 79$

2. Form the third linear equation from the condition involving the final 25th question.

If the 25th question is answered wrongly, the number of wrong questions becomes $c + 1$. The total number of correct and partially correct answers remains $a + b$.
The condition states the total of correct and partially correct questions is four times the wrong questions:
$a + b = 4(c + 1)$

3. Solve the system of equations for $c$.

From the first equation, we can express $a + b$ as:
$a + b = 24 - c$

Substitute this expression into the third equation:
$24 - c = 4(c + 1)$
$24 - c = 4c + 4$
$5c = 20 \implies c = 4$

4. Solve the remaining equations to find the values of $a$ and $b$.

Substitute $c = 4$ into the first two equations to form a 2-variable system:
$a + b + 4 = 24 \implies a + b = 20 \quad \text{--- (1)}$
$5a + 2b - 3(4) = 79 \implies 5a + 2b = 91 \quad \text{--- (2)}$

Multiply equation (1) by 2:
$2a + 2b = 40 \quad \text{--- (3)}$

Subtract equation (3) from equation (2):
$(5a + 2b) - (2a + 2b) = 91 - 40$
$3a = 51 \implies a = 17$

Substitute $a = 17$ back into equation (1):
$17 + b = 20 \implies b = 3$