To solve the equation \( an + bn = cn \), where \( a \), \( b \), and \( c \) are coefficients and \( n \) is a variable, we need to express the variable \( n \) in terms of \( a \), \( b \), and \( c \). Here’s a step-by-step solution:

1. **Combine like terms involving \( n \):**

   Since all terms involve \( n \), we can factor it out:

   \[
   an + bn = cn
   \]

   \[
   (a + b)n = cn
   \]

2. **Isolate \( n \):**

   To solve for \( n \), we can either:

   - Divide both sides by \( n \), but this only works if \( n \neq 0 \).
   - Rearrange the equation to isolate \( n \).

   Assuming \( n \neq 0 \), divide both sides by \( n \):

   \[
   a + b = c
   \]

   This equation suggests that \( n \) can take any value as long as \( a + b = c \). If \( a + b \neq c \), then there is no solution for non-zero \( n \).

3. **Special case when \( n = 0 \):**

   If \( n = 0 \), then the equation holds true regardless of the values of \( a \), \( b \), and \( c \), because:

   \[
   an + bn = cn \rightarrow 0 + 0 = 0
   \]

Therefore, the solution depends on the values of \( a \), \( b \), and \( c \):

- If \( a + b = c \), then \( n \) can be any real number.
- If \( a + b \neq c \), the equation only holds if \( n = 0 \).

**Conclusion:**

The equation \( an + bn = cn \) can be solved as follows:

- If \( a + b = c \), then \( n \) can take any value (including 0).
- If \( a + b \neq c \), then the only solution is \( n = 0 \).

Do you want more details or have any questions about the solution? Here are 5 related questions you might find interesting:

1. What if the equation was \( an^2 + bn = cn \)? How would that change the solution?
2. How can this problem be approached if \( a \), \( b \), and \( c \) are functions of \( n \)?
3. What if the coefficients \( a \), \( b \), and \( c \) include complex numbers? How does that affect the solution?
4. Can you solve for \( n \) if additional constraints are given, such as \( n > 0 \)?
5. How does the solution change if the equation is in terms of a different variable, say \( x \), instead of \( n \)?

**Tip:** Always check for special cases in algebraic equations, such as \( n = 0 \) or \( n \) causing division by zero, to ensure your solution is comprehensive and correct.

Learn how to solve the linear equation an + bn = cn with a detailed step-by-step solution. Understand the conditions under which the equation has solutions and explore related algebraic concepts. Suitable for students in Grades 7-9.

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