https://file.aiquickdraw.com/mathbot/file/1727316114457sk6z8zri.jpg

The equation provided in the image is:

\[
12c - 3 + 14 - 9c = 1 + 2c + 10 + c
\]

### Step-by-step solution:

1. **Simplify both sides**:
   - Left-hand side (LHS): 
     \[
     12c - 9c - 3 + 14 = 3c + 11
     \]
   - Right-hand side (RHS): 
     \[
     1 + 10 + 2c + c = 11 + 3c
     \]

   Now the equation becomes:
   \[
   3c + 11 = 11 + 3c
   \]

2. **Subtract \(3c\) from both sides**:
   \[
   11 = 11
   \]

   This is a true statement, meaning the equation holds for all values of \(c\).

### Conclusion:
Since the equation simplifies to a true statement, it means that **\(c\) can be any real number**. This is an identity.

---

Would you like more details or explanations about this process? Here are a few related questions to explore:

1. What is an identity equation in algebra?
2. How can we recognize when an equation has infinite solutions?
3. How would this problem change if the equation didn't simplify to a true statement?
4. What is the difference between equations with no solution, one solution, and infinite solutions?
5. How can similar linear equations appear in real-world problems?

**Tip**: Always simplify both sides of an equation first before solving; it makes it easier to identify patterns like identities or contradictions.

Solve for c: 12c - 3 + 14 - 9c = 1 + 2c + 10 + c

Explore how to solve the identity equation 12c - 3 + 14 - 9c = 1 + 2c + 10 + c with a detailed step-by-step solution. Learn about algebraic simplification, identity equations, and the properties of linear equations. Ideal for students in Grades 8-10.

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