Let's solve the equation \( 7c + 12 = -4c + 78 \).

### Step 1: Move all terms involving \( c \) to one side
Add \( 4c \) to both sides:
\[
7c + 4c + 12 = 78
\]
This simplifies to:
\[
11c + 12 = 78
\]

### Step 2: Move constant terms to the other side
Subtract 12 from both sides:
\[
11c = 78 - 12
\]
\[
11c = 66
\]

### Step 3: Solve for \( c \)
Divide both sides by 11:
\[
c = \frac{66}{11} = 6
\]

### Final Answer:
\[
c = 6
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What happens if we change the constant in the equation?
2. How would the solution change if one term had a squared variable, like \( 7c^2 + 12 \)?
3. What are the steps to solving equations with fractions, like \( \frac{7}{2}c + 12 = -4c + 78 \)?
4. How can you check your solution after solving the equation?
5. What happens when you have equations with no solutions or infinite solutions?

**Tip:** Always check your solution by substituting it back into the original equation to verify if both sides are equal!

Learn how to solve the linear equation 7c + 12 = -4c + 78 with a detailed step-by-step solution. This problem covers key algebraic concepts such as moving variables to one side and solving for a single variable. Suitable for students in Grades 6-8.

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