https://file.aiquickdraw.com/mathbot/file/1734292768573xqffxc6s.jpg

The problem states:

1. Fred is **4 times** as old as Nathan.  
2. Fred is also **27 years older** than Nathan.

We are asked to identify the correct system of equations, where:  
- \( f \) = Fred's age  
- \( n \) = Nathan's age  

---

### Translating the statements into equations:
1. "Fred is 4 times as old as Nathan" gives:  
   \[
   f = 4n
   \]

2. "Fred is 27 years older than Nathan" gives:  
   \[
   f = n + 27
   \]

---

### Checking Option A:
Option A provides the following system of equations:  
\[
\begin{cases}
n = 4f \\
n = f + 27
\end{cases}
\]

- In the first equation, **\( n = 4f \)** is **incorrect** since the problem says Fred (\( f \)) is 4 times Nathan (\( n \)), not the other way around.  
- The second equation \( n = f + 27 \) is also **incorrect** because it suggests Nathan is older than Fred by 27 years, which is not true.

Thus, **Option A is not correct**.

---

The correct system of equations should be:  
\[
\begin{cases}
f = 4n \\
f = n + 27
\end{cases}
\]

---

### Final Note:
Let me know if you'd like me to solve for Fred's and Nathan's ages, or if you have any additional questions!

---

### Related Questions:
1. How can you solve a system of two linear equations algebraically?  
2. What are the methods for solving systems of equations (e.g., substitution, elimination)?  
3. What happens when a system of equations has no solution?  
4. Can you graph these equations to find the intersection point?  
5. How can you verify the solution of a system of equations?

---

### Tip:
Always carefully identify the relationships in word problems and assign variables logically before forming equations.

Fred is 4 times as old as Nathan and is also 27 years older than Nathan. Let f be Fred's age and let n be Nathan's age. Which system of equations represents this situation?

Solve the system of equations where Fred is 4 times as old as Nathan and 27 years older. This problem explores linear relationships and systems of equations, providing essential practice for students in Grades 6-8.

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