Find the equation of the line parallel to x - axis passing through the intersection of the lines `3ax+2by+7b=0` and `3bx-2ay-7a=0`, where `(a, b) ne (0,0)`.

Answer

To find the equation of the line parallel to the x-axis passing through the intersection of the lines \(3ax + 2by + 7b = 0\) and \(3bx - 2ay - 7a = 0\), we can follow these steps:

### Step 1: Find the intersection point of the two lines.

We have two equations:
1. \(3ax + 2by + 7b = 0\) (Equation 1)
2. \(3bx - 2ay - 7a = 0\) (Equation 2)

To find the intersection, we can solve these equations simultaneously.

### Step 2: Multiply the equations to eliminate one variable.

Multiply Equation 1 by \(b\) and Equation 2 by \(a\):
- \(b(3ax + 2by + 7b) = 0 \Rightarrow 3abx + 2b^2y + 7b^2 = 0\) (Equation 3)
- \(a(3bx - 2ay - 7a) = 0 \Rightarrow 3abx - 2a^2y - 7a^2 = 0\) (Equation 4)

### Step 3: Subtract the two equations.

Now, subtract Equation 4 from Equation 3:
\[ (3abx + 2b^2y + 7b^2) - (3abx - 2a^2y - 7a^2) = 0 \]
This simplifies to:
\[ 2b^2y + 7b^2 + 2a^2y + 7a^2 = 0 \]
Combining like terms gives:
\[ (2b^2 + 2a^2)y + 7(b^2 + a^2) = 0 \]

### Step 4: Solve for \(y\).

Rearranging the equation:
\[ (2b^2 + 2a^2)y = -7(b^2 + a^2) \]
Dividing both sides by \(2(a^2 + b^2)\) (which is non-zero since \(a\) and \(b\) are not both zero):
\[ y = -\frac{7}{2} \]

### Step 5: Substitute \(y\) back to find \(x\).

Now, substitute \(y = -\frac{7}{2}\) back into either Equation 1 or Equation 2. We will use Equation 1:
\[ 3ax + 2b\left(-\frac{7}{2}\right) + 7b = 0 \]
This simplifies to:
\[ 3ax - 7b + 7b = 0 \Rightarrow 3ax = 0 \]
Thus, we find:
\[ x = 0 \]

### Step 6: Identify the intersection point.

The intersection point of the two lines is:
\[ (0, -\frac{7}{2}) \]

### Step 7: Write the equation of the line parallel to the x-axis.

A line parallel to the x-axis has the equation of the form \(y = c\). Since our intersection point has a \(y\)-coordinate of \(-\frac{7}{2}\), the equation of the line is:
\[ y = -\frac{7}{2} \]

### Final Answer:
The equation of the line parallel to the x-axis passing through the intersection of the given lines is:
\[ y + \frac{7}{2} = 0 \]