Abby attempted a mathematics quiz with 80 questions.

For each incorrect answer, 3 marks were deducted.

For each correct answer, 5 marks were awarded.

Abby's score was 312.

How many questions did Abby answer correctly?

Let \( x \) be the number of questions Abby answered correctly, and \( y \) be the number of questions she answered incorrectly.

We are given the following information:

1. The total number of questions is 80:
   \[
   x + y = 80
   \]

2. Abby's score is 312, where she earns 5 marks for each correct answer and loses 3 marks for each incorrect answer:
   \[
   5x - 3y = 312
   \]

We now have a system of two equations:

\[
\begin{aligned}
1) & \quad x + y = 80 \\
2) & \quad 5x - 3y = 312
\end{aligned}
\]

### Step 1: Solve the first equation for \( y \)
\[
y = 80 - x
\]

### Step 2: Substitute \( y = 80 - x \) into the second equation
\[
5x - 3(80 - x) = 312
\]

### Step 3: Simplify and solve for \( x \)
\[
5x - 240 + 3x = 312
\]
\[
8x - 240 = 312
\]
\[
8x = 552
\]
\[
x = \frac{552}{8} = 69
\]

### Step 4: Calculate \( y \)
\[
y = 80 - 69 = 11
\]

So, Abby answered **69 questions correctly**.

Would you like further details or have any other questions?

Here are 5 related questions:
1. What would Abby's score be if she answered all questions correctly?
2. How many incorrect answers would lead to a score of zero?
3. If 10 additional questions were added to the quiz, how would this affect Abby's score?
4. How would Abby's score change if the deduction per incorrect answer increased to 4 marks?
5. What would be the minimum number of correct answers needed for a positive score?

**Tip:** When solving systems of equations, substitution is useful when one equation can be easily solved for one variable.

Explore how to solve a system of equations to determine how many questions Abby answered correctly on her mathematics quiz. This problem involves understanding systems of linear equations and applying algebraic techniques. Suitable for Grades 9-12.

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