The problem states that for a math exam, each correct answer is worth 4 points, and each incorrect answer results in a deduction of 1 point. A student answered 23 questions in total and scored 72 points. We are tasked to find how many incorrect answers the student gave by setting up and solving a system of equations.

Solution:

Let:

• $x$: number of correct answers,
• $y$: number of incorrect answers.

From the problem, we have the following equations:

1. $x + y = 23$ (since the total number of questions answered is 23).
2. $4x - y = 72$ (the total score formula, where each correct answer adds 4 points, and each incorrect answer deducts 1 point).

Step 1: Solve the system of equations.

From Equation (1): $x = 23 - y$

Substitute $x = 23 - y$ into Equation (2): $4(23 - y) - y = 72$ Simplify: $92 - 4y - y = 72$ $92 - 5y = 72$ $5y = 20$ $y = 4$

Step 2: Find $x$.

Using $x = 23 - y$: $x = 23 - 4 = 19$

Final Answer:

The student gave 4 incorrect answers.

Would you like further clarification on the equations? Here are 5 related questions:

1. How would the solution change if the deduction for incorrect answers was 2 points instead of 1?
2. Can you explain the role of substitution in solving this system of equations?
3. What happens if the student answers fewer than 23 questions in this scenario?
4. How would the equations look if there were also unanswered questions?
5. Can this problem be solved graphically, and how?

Tip: Always double-check your substitution step to avoid small arithmetic errors when solving systems of equations!