In the finals of a General Knowledge Quiz, a team is required to answer 25 questions. Each question that is correctly answered scores 5 points, while a question that is wrongly answered is deducted 3 points. If the answer is partially correct, the team scores 2 points.
After 24 questions, the results are shown in the following table. | Number of Questions | | |
| :---: | :---: | :---: | :---: |
| Correct | Partially Correct | Wrong | Points |
| $a$ | $b$ | $c$ | 79 | If the team answers the last question wrongly, then the total number of questions answered correctly and partially correct is four times the number of questions answered wrongly. By forming a system of linear equations, find the values of $a$, $b$ and $c$.

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):
$17 + b = 20 \implies b = 3$