Solve the following equations.Check your result in each case
`1/2X-3=5+1/3X`

Answer

To solve the equation \( \frac{1}{2}X - 3 = 5 + \frac{1}{3}X \), we will follow these steps:

### Step 1: Move all terms involving \( X \) to one side and constant terms to the other side.
We start with the equation:
\[ \frac{1}{2}X - 3 = 5 + \frac{1}{3}X \]
We can move \( \frac{1}{3}X \) to the left side by subtracting it from both sides:
\[ \frac{1}{2}X - \frac{1}{3}X - 3 = 5 \]

### Step 2: Simplify the left side.
To combine \( \frac{1}{2}X \) and \( \frac{1}{3}X \), we need a common denominator. The least common multiple of 2 and 3 is 6. We can rewrite the fractions:
\[ \frac{1}{2}X = \frac{3}{6}X \quad \text{and} \quad \frac{1}{3}X = \frac{2}{6}X \]
Now substituting these back into the equation gives:
\[ \left(\frac{3}{6}X - \frac{2}{6}X\right) - 3 = 5 \]
This simplifies to:
\[ \frac{1}{6}X - 3 = 5 \]

### Step 3: Isolate \( X \).
Now, we will add 3 to both sides to isolate the term with \( X \):
\[ \frac{1}{6}X = 5 + 3 \]
This simplifies to:
\[ \frac{1}{6}X = 8 \]

### Step 4: Solve for \( X \).
To solve for \( X \), we multiply both sides by 6:
\[ X = 8 \times 6 \]
Calculating this gives:
\[ X = 48 \]

### Step 5: Check the solution.
Now, we will substitute \( X = 48 \) back into the original equation to verify:
\[ \frac{1}{2}(48) - 3 = 5 + \frac{1}{3}(48) \]
Calculating the left side:
\[ \frac{1}{2}(48) = 24 \quad \Rightarrow \quad 24 - 3 = 21 \]
Calculating the right side:
\[ \frac{1}{3}(48) = 16 \quad \Rightarrow \quad 5 + 16 = 21 \]
Since both sides are equal:
\[ 21 = 21 \]
Thus, the solution is verified.

### Final Answer:
\[ X = 48 \]