Solve `: (2)/(x) +(2)/(3y ) = (1)/(6) and (3)/(x) + (2)/(y) =0 . `
Hence , find 'm' for which y = mx-4 .

Answer

To solve the equations \( \frac{2}{x} + \frac{2}{3y} = \frac{1}{6} \) and \( \frac{3}{x} + \frac{2}{y} = 0 \) simultaneously, and then find the value of \( m \) such that \( y = mx - 4 \), we can follow these steps:

### Step 1: Solve the second equation for \( y \)
From the second equation:
\[ \frac{3}{x} + \frac{2}{y} = 0 \]
Rearranging gives:
\[ \frac{2}{y} = -\frac{3}{x} \]
Multiplying both sides by \( y \) and \( x \):
\[ 2x = -3y \]
Thus, we can express \( y \) in terms of \( x \):
\[ y = -\frac{2}{3}x \]

### Step 2: Substitute \( y \) into the first equation
Now substitute \( y = -\frac{2}{3}x \) into the first equation:
\[ \frac{2}{x} + \frac{2}{3(-\frac{2}{3}x)} = \frac{1}{6} \]
This simplifies to:
\[ \frac{2}{x} - \frac{1}{3x} = \frac{1}{6} \]

### Step 3: Combine the fractions
To combine \( \frac{2}{x} - \frac{1}{3x} \), we need a common denominator:
\[ \frac{6}{3x} - \frac{1}{3x} = \frac{5}{3x} \]
So we have:
\[ \frac{5}{3x} = \frac{1}{6} \]

### Step 4: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 5 \cdot 6 = 3x \implies 30 = 3x \implies x = 10 \]

### Step 5: Find \( y \) using \( x \)
Now substitute \( x = 10 \) back into the equation for \( y \):
\[ y = -\frac{2}{3}(10) = -\frac{20}{3} \]

### Step 6: Find \( m \) using \( y = mx - 4 \)
Now we have \( y = -\frac{20}{3} \) and \( x = 10 \). Substitute these into the equation \( y = mx - 4 \):
\[ -\frac{20}{3} = m(10) - 4 \]
Rearranging gives:
\[ m(10) = -\frac{20}{3} + 4 \]
Convert 4 to a fraction:
\[ 4 = \frac{12}{3} \]
Thus:
\[ m(10) = -\frac{20}{3} + \frac{12}{3} = -\frac{8}{3} \]
Now, divide both sides by 10:
\[ m = -\frac{8}{3} \cdot \frac{1}{10} = -\frac{8}{30} = -\frac{4}{15} \]

### Final Result
The value of \( m \) is:
\[ m = -\frac{4}{15} \]