Solve ` : 4x + ( 6)/( y) = 15 and 6x - ( 8)/( y) = 14. `
Hence , find the value of ' k ' , if `y= kx - 2 `

Answer

To solve the equations \(4x + \frac{6}{y} = 15\) and \(6x - \frac{8}{y} = 14\), and find the value of \(k\) in the equation \(y = kx - 2\), we can follow these steps:

### Step 1: Substitute \( \frac{1}{y} \) with a new variable
Let \( t = \frac{1}{y} \). Then we can rewrite the equations:
- The first equation becomes \( 4x + 6t = 15 \).
- The second equation becomes \( 6x - 8t = 14 \).

### Step 2: Rearrange the equations
From the first equation:
\[ 6t = 15 - 4x \quad \Rightarrow \quad t = \frac{15 - 4x}{6} \]
From the second equation:
\[ -8t = 14 - 6x \quad \Rightarrow \quad t = \frac{6x - 14}{8} \]

### Step 3: Set the two expressions for \(t\) equal to each other
\[ \frac{15 - 4x}{6} = \frac{6x - 14}{8} \]

### Step 4: Cross-multiply to eliminate the fractions
\[ 8(15 - 4x) = 6(6x - 14) \]
Expanding both sides:
\[ 120 - 32x = 36x - 84 \]

### Step 5: Combine like terms
Bringing all \(x\) terms to one side and constant terms to the other:
\[ 120 + 84 = 36x + 32x \]
\[ 204 = 68x \]

### Step 6: Solve for \(x\)
\[ x = \frac{204}{68} = 3 \]

### Step 7: Substitute \(x\) back to find \(t\)
Using \(x = 3\) in the first equation for \(t\):
\[ t = \frac{15 - 4(3)}{6} = \frac{15 - 12}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 8: Find \(y\) using \(t\)
Since \(t = \frac{1}{y}\), we have:
\[ \frac{1}{y} = \frac{1}{2} \quad \Rightarrow \quad y = 2 \]

### Step 9: Find \(k\) using \(y = kx - 2\)
Substituting \(y = 2\) and \(x = 3\) into the equation:
\[ 2 = k(3) - 2 \]
Adding 2 to both sides:
\[ 4 = 3k \]
Dividing by 3:
\[ k = \frac{4}{3} \]

### Final Answer
The value of \(k\) is \(\frac{4}{3}\).