To solve the equation \( \frac{X}{3} + \frac{x - 1}{4} = 2 + x \), let's follow these steps:

### Step 1: Eliminate the Fractions
To eliminate the fractions, find the least common denominator (LCD) of 3 and 4, which is 12. Multiply every term in the equation by 12:

\[
12 \times \left(\frac{X}{3}\right) + 12 \times \left(\frac{x - 1}{4}\right) = 12 \times 2 + 12 \times x
\]

This simplifies to:

\[
4X + 3(x - 1) = 24 + 12x
\]

### Step 2: Distribute and Simplify
Now, distribute the 3 across the \( (x - 1) \):

\[
4X + 3x - 3 = 24 + 12x
\]

### Step 3: Combine Like Terms
Next, combine like terms and move all terms involving \( x \) to one side of the equation:

\[
4X + 3x - 12x = 24 + 3
\]

Simplify further:

\[
4X - 9x = 27
\]

### Step 4: Factor and Solve for \( x \)
Factor out the common factor from the left side:

\[
x(4 - 9) = 27
\]

Simplify:

\[
-5x = 27
\]

Finally, solve for \( x \):

\[
x = -\frac{27}{5}
\]

So, \( x = -5.4 \).

Would you like further details or have any questions about the solution?

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Here are some related questions for practice:
1. Solve \( \frac{y}{2} + \frac{y + 3}{5} = 4 + y \).
2. If \( x = -5.4 \), what is \( x^2 + 3x + 2 \)?
3. Solve the equation \( 3x - 7 = 2x + 5 \).
4. What is the solution to \( \frac{z}{4} + \frac{z - 2}{3} = z + 1 \)?
5. Solve for \( y \) in the equation \( 2y - \frac{y}{2} = 5 + \frac{y}{4} \).

**Tip:** Always double-check each step when solving equations, especially when working with fractions and distributing terms.

Learn how to solve the equation X/3 + (x - 1)/4 = 2 + x with detailed step-by-step solutions. This problem covers equations with fractions, requiring knowledge of finding LCD (Least Common Denominator) and simplifying algebraic expressions. Suitable for high school students.

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