Let's solve the equation:

\[
-12 + \frac{1}{4}x + 40 = 35
\]

### Step 1: Combine like terms.
First, simplify the constants on the left-hand side:

\[
-12 + 40 = 28
\]

This simplifies the equation to:

\[
28 + \frac{1}{4}x = 35
\]

### Step 2: Subtract 28 from both sides.
Now, subtract 28 from both sides to isolate the term with \(x\):

\[
\frac{1}{4}x = 35 - 28
\]

\[
\frac{1}{4}x = 7
\]

### Step 3: Solve for \(x\).
To eliminate the fraction, multiply both sides of the equation by 4:

\[
x = 7 \times 4
\]

\[
x = 28
\]

### Final Answer:
The solution is \(x = 28\).

Would you like further clarification or details on any step?

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Here are some related questions to explore:
1. How would the equation change if the \( \frac{1}{4}x \) term had a negative sign?
2. Can you solve a similar equation: \( -5 + \frac{1}{2}x + 15 = 10 \)?
3. What happens if you multiply or divide both sides of an equation by a negative number?
4. How do you check if your solution to an equation is correct?
5. How would this equation be solved if it involved square roots?

**Tip:** Always simplify an equation as much as possible before solving for the variable. This often makes the process much easier!

Explore how to solve the linear equation -12 + 1/4x + 40 = 35. This problem covers fundamental algebra concepts, including combining like terms and solving for a variable. Suitable for students in Grades 6-8.

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