For every natural number x and y the value of x-y when `y+7/x+y/13=6 3/143` is :

Options

A. 5
B. 6
C. 2
D. not unique

Answer

To solve the equation given in the problem, we will follow these steps:

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ \frac{y + 7}{x} + \frac{y}{13} = 6 \frac{3}{143} \]
First, convert the mixed fraction \(6 \frac{3}{143}\) into an improper fraction:
\[ 6 \frac{3}{143} = \frac{6 \times 143 + 3}{143} = \frac{858 + 3}{143} = \frac{861}{143} \]
Thus, our equation becomes:
\[ \frac{y + 7}{x} + \frac{y}{13} = \frac{861}{143} \]

### Step 2: Find a common denominator
The common denominator for the left side of the equation is \(13x\). Rewrite the equation:
\[ \frac{(y + 7) \cdot 13 + y \cdot x}{13x} = \frac{861}{143} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ 143 \left( (y + 7) \cdot 13 + y \cdot x \right) = 861 \cdot 13x \]

### Step 4: Simplify the equation
Distributing on the left side:
\[ 143 \cdot 13y + 143 \cdot 91 + 143xy = 11193x \]
This simplifies to:
\[ 143 \cdot 13y + 143xy + 11753 = 11193x \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 143xy + 143 \cdot 13y - 11193x + 11753 = 0 \]

### Step 6: Substitute values
To find natural numbers \(x\) and \(y\), we can try substituting \(x = 11\) (as suggested in the video):
\[ 143 \cdot 11y + 143 \cdot 13y - 11193 \cdot 11 + 11753 = 0 \]

### Step 7: Solve for \(y\)
Substituting \(x = 11\):
\[ 143 \cdot 11y + 143 \cdot 13y - 123123 + 11753 = 0 \]
This simplifies to:
\[ 154y + 91 = 861 \]
Subtracting 91 from both sides:
\[ 154y = 770 \]
Dividing by 154:
\[ y = 5 \]

### Step 8: Calculate \(x - y\)
Now we find \(x - y\):
\[ x - y = 11 - 5 = 6 \]

### Final Answer
The value of \(x - y\) is:
\[ \boxed{6} \]