\(\frac{1}{2}x+2=3\frac{1}{4}\cdot 4x-3\)

Let's solve the equation \(\frac{1}{2}x + 2 = 3\frac{1}{4} \cdot 4x - 3\).

### Step 1: Convert the mixed fraction to an improper fraction
First, convert the mixed number \(3\frac{1}{4}\) into an improper fraction:

\[
3\frac{1}{4} = \frac{13}{4}
\]

So the equation becomes:

\[
\frac{1}{2}x + 2 = \frac{13}{4} \cdot 4x - 3
\]

### Step 2: Simplify the equation
Next, multiply \(\frac{13}{4}\) by \(4x\):

\[
\frac{13}{4} \cdot 4x = 13x
\]

Now, the equation is:

\[
\frac{1}{2}x + 2 = 13x - 3
\]

### Step 3: Eliminate the fraction
To eliminate the fraction, multiply every term in the equation by 2:

\[
2 \cdot \left(\frac{1}{2}x + 2\right) = 2 \cdot \left(13x - 3\right)
\]

This simplifies to:

\[
x + 4 = 26x - 6
\]

### Step 4: Solve for \(x\)
Subtract \(x\) from both sides:

\[
4 = 25x - 6
\]

Add 6 to both sides:

\[
10 = 25x
\]

Divide by 25:

\[
x = \frac{10}{25} = \frac{2}{5}
\]

### Final Answer
The solution to the equation is:

\[
x = \frac{2}{5}
\]

Would you like a detailed explanation of any step? 

---

Here are some related questions you might find interesting:

1. How do you solve equations involving multiple fractions?
2. What is the general method for solving linear equations?
3. How do you handle mixed fractions in algebraic equations?
4. Can you show how to solve an equation with variables on both sides?
5. How do you solve equations involving decimals?
6. What happens if a fraction has a variable in both the numerator and denominator?
7. How do you check the solution to an equation?
8. What are some common mistakes to avoid when solving linear equations?

**Tip:** Always double-check your solution by substituting it back into the original equation.

Learn how to solve the equation involving fractions: 1/2x + 2 = 3 1/4 * 4x - 3. Detailed step-by-step solution provided. Suitable for Grades 7-9.

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