Solution of the system of linear equations `(5/x)-(4/y)=3` and `(9/x)-(8/y)=7` is?

Answer

To solve the system of linear equations given by

\[ \frac{5}{x} - \frac{4}{y} = 3 \]

and

\[ \frac{9}{x} - \frac{8}{y} = 7, \]

we can start by substituting \( \frac{1}{x} \) as \( a \) and \( \frac{1}{y} \) as \( b \). This will simplify our equations.

### Step 1: Substitute variables
Let:
\[ a = \frac{1}{x} \quad \text{and} \quad b = \frac{1}{y}. \]

Now, we can rewrite the equations as:
\[ 5a - 4b = 3 \quad \text{(1)} \]
\[ 9a - 8b = 7 \quad \text{(2)} \]

### Step 2: Multiply equation (1)
To eliminate \( b \), we can multiply equation (1) by 2:
\[ 2(5a - 4b) = 2(3) \implies 10a - 8b = 6 \quad \text{(3)} \]

### Step 3: Subtract equation (2) from equation (3)
Now, we will subtract equation (2) from equation (3):
\[ (10a - 8b) - (9a - 8b) = 6 - 7 \]
This simplifies to:
\[ 10a - 9a = -1 \implies a = -1. \]

### Step 4: Substitute \( a \) back to find \( b \)
Now that we have \( a \), we can substitute it back into equation (1) to find \( b \):
\[ 5(-1) - 4b = 3 \implies -5 - 4b = 3. \]
Rearranging gives:
\[ -4b = 3 + 5 \implies -4b = 8 \implies b = -2. \]

### Step 5: Find \( x \) and \( y \)
Now we can find \( x \) and \( y \) using the values of \( a \) and \( b \):
\[ a = \frac{1}{x} \implies -1 = \frac{1}{x} \implies x = -1, \]
\[ b = \frac{1}{y} \implies -2 = \frac{1}{y} \implies y = -\frac{1}{2}. \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = -1 \quad \text{and} \quad y = -\frac{1}{2}. \]