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We are given the equation \( p + 3m = n \) and asked to find a possible value of \( m + n - p \), with \( m \), \( n \), and \( p \) being integers.

Let’s begin by expressing \( n \) in terms of \( m \) and \( p \):
\[
n = p + 3m
\]
Now, substitute this into the expression \( m + n - p \):
\[
m + n - p = m + (p + 3m) - p
\]
Simplify the expression:
\[
m + n - p = m + p + 3m - p = 4m
\]
Thus, the expression simplifies to:
\[
m + n - p = 4m
\]
Now, we need to determine which of the given options (10, 18, 24, 34, 42) could be equal to \( 4m \).

Since \( m + n - p = 4m \), the value of \( 4m \) must be one of the given options. The integer value of \( m \) that satisfies \( 4m \) is:

- \( 4m = 10 \), so \( m = 2.5 \) (not an integer)
- \( 4m = 18 \), so \( m = 4.5 \) (not an integer)
- \( 4m = 24 \), so \( m = 6 \) (integer)
- \( 4m = 34 \), so \( m = 8.5 \) (not an integer)
- \( 4m = 42 \), so \( m = 10.5 \) (not an integer)

Therefore, the only valid value for \( m \) is 6, which gives us \( m + n - p = 24 \).

Thus, the correct answer is **C) 24**.

Do you need further details or have any questions?

Here are five related questions to expand your understanding:
1. How would the equation change if the relationship between \( p \) and \( n \) was \( p + 2m = n \)?
2. Can you solve for \( p \) if \( m = 6 \) and \( n = 24 \)?
3. What if the equation were \( p + 4m = n \), what would be the new expression for \( m + n - p \)?
4. How would you solve the problem if only \( m + n + p \) was given?
5. What are the properties of integers used to solve this equation?

**Tip:** When simplifying expressions, always substitute known relationships to reduce complexity and reveal patterns.

m, n, p birer tam sayıdır. p + 3m = n olduğuna göre, m + n - p değeri hangisi olabilir? A) 10 B) 18 C) 24 D) 34 E) 42

Solve the algebraic equation p + 3m = n and determine the value of m + n - p. This step-by-step guide explores integer operations and algebraic simplification for students in Grades 7-9.

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