Solve :
`(9)/(x) - (4)/(y) = 8`
`(13)/(x) + (7)/(y) =101`

Answer

To solve the simultaneous equations:

1. \(\frac{9}{x} - \frac{4}{y} = 8\)
2. \(\frac{13}{x} + \frac{7}{y} = 101\)

we can simplify the problem by substituting \( \frac{1}{x} = m \) and \( \frac{1}{y} = n \). This gives us:

1. \(9m - 4n = 8\) (Equation 1)
2. \(13m + 7n = 101\) (Equation 2)

### Step 1: Rewrite the equations
We rewrite the equations using our substitutions:
- \(9m - 4n = 8\)
- \(13m + 7n = 101\)

### Step 2: Eliminate one variable
To eliminate \(n\), we can multiply Equation 1 by 7 and Equation 2 by 4 to make the coefficients of \(n\) the same:

- Multiply Equation 1 by 7:
\[ 7(9m - 4n) = 7(8) \implies 63m - 28n = 56 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 4:
\[ 4(13m + 7n) = 4(101) \implies 52m + 28n = 404 \quad \text{(Equation 4)} \]

### Step 3: Add the equations
Now, we add Equation 3 and Equation 4:
\[ (63m - 28n) + (52m + 28n) = 56 + 404 \]
This simplifies to:
\[ 115m = 460 \]

### Step 4: Solve for \(m\)
Now, we can solve for \(m\):
\[ m = \frac{460}{115} = 4 \]

### Step 5: Substitute back to find \(n\)
Now that we have \(m\), we substitute \(m = 4\) back into Equation 1 to find \(n\):
\[ 9(4) - 4n = 8 \]
This simplifies to:
\[ 36 - 4n = 8 \]
Rearranging gives:
\[ -4n = 8 - 36 \implies -4n = -28 \implies n = 7 \]

### Step 6: Find \(x\) and \(y\)
Now that we have \(m\) and \(n\):
- Since \(m = \frac{1}{x}\), we have:
\[ x = \frac{1}{m} = \frac{1}{4} \]

- Since \(n = \frac{1}{y}\), we have:
\[ y = \frac{1}{n} = \frac{1}{7} \]

### Final Answer
Thus, the solution to the simultaneous equations is:
\[ x = \frac{1}{4}, \quad y = \frac{1}{7} \]