Solve the following pair of equations:
`4x+ (6)/(y)=15, 6x- (8)/(y)= 14, y ne 0`

Answer

To solve the pair of equations given by:

1. \( 4x + \frac{6}{y} = 15 \) (Equation 1)
2. \( 6x - \frac{8}{y} = 14 \) (Equation 2)

we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply both equations by \( y \) (since \( y \neq 0 \)).

From Equation 1:
\[ y(4x) + y\left(\frac{6}{y}\right) = 15y \]
This simplifies to:
\[ 4xy + 6 = 15y \quad \text{(Equation 3)} \]

From Equation 2:
\[ y(6x) - y\left(\frac{8}{y}\right) = 14y \]
This simplifies to:
\[ 6xy - 8 = 14y \quad \text{(Equation 4)} \]

### Step 2: Rearranging the equations
Now, we can rearrange both equations to isolate terms involving \( y \):

From Equation 3:
\[ 4xy - 15y + 6 = 0 \quad \text{(Rearranged Equation 3)} \]

From Equation 4:
\[ 6xy - 14y - 8 = 0 \quad \text{(Rearranged Equation 4)} \]

### Step 3: Solve the equations simultaneously
Now we have two equations:
1. \( 4xy - 15y + 6 = 0 \)
2. \( 6xy - 14y - 8 = 0 \)

We can express \( y \) in terms of \( x \) from one of the equations. Let's solve for \( y \) in terms of \( x \) from Equation 3.

Rearranging Equation 3 gives:
\[ 4xy - 15y = -6 \]
Factoring out \( y \):
\[ y(4x - 15) = -6 \]
Thus,
\[ y = \frac{-6}{4x - 15} \quad \text{(Equation 5)} \]

### Step 4: Substitute \( y \) into the other equation
Now substitute Equation 5 into Equation 4:
\[ 6x\left(\frac{-6}{4x - 15}\right) - 14\left(\frac{-6}{4x - 15}\right) - 8 = 0 \]
Multiplying through by \( (4x - 15) \) to eliminate the denominator gives:
\[ -36x + 84 - 8(4x - 15) = 0 \]
Expanding:
\[ -36x + 84 - 32x + 120 = 0 \]
Combining like terms:
\[ -68x + 204 = 0 \]
Solving for \( x \):
\[ 68x = 204 \implies x = \frac{204}{68} = 3 \]

### Step 5: Substitute \( x \) back to find \( y \)
Now substitute \( x = 3 \) back into Equation 5 to find \( y \):
\[ y = \frac{-6}{4(3) - 15} = \frac{-6}{12 - 15} = \frac{-6}{-3} = 2 \]

### Final Solution
Thus, the solution to the pair of equations is:
\[ x = 3, \quad y = 2 \]