Equation of a line passing through the intersection of the lines `7x-y+2=0` and `x-3y+6=0` parallel to x - axis is

Options

A. y = 2
B. `y=-2`
C. y = 3
D. `y=-3`

Answer

To find the equation of a line that passes through the intersection of the lines \(7x - y + 2 = 0\) and \(x - 3y + 6 = 0\) and is parallel to the x-axis, we can follow these steps:

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

We have two equations:
1. \(7x - y + 2 = 0\) (Equation 1)
2. \(x - 3y + 6 = 0\) (Equation 2)

We can solve these equations simultaneously to find the intersection point.

From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 7x + 2 \]

Now, substitute this expression for \(y\) into Equation 2:
\[ x - 3(7x + 2) + 6 = 0 \]
Expanding this gives:
\[ x - 21x - 6 + 6 = 0 \]
Simplifying, we have:
\[ -20x = 0 \implies x = 0 \]

Now, substitute \(x = 0\) back into the expression for \(y\):
\[ y = 7(0) + 2 = 2 \]

Thus, the intersection point is \((0, 2)\).

### Step 2: Determine the equation of the line parallel to the x-axis.

A line that is parallel to the x-axis has a slope of 0. The general equation of a line with slope \(m\) passing through a point \((x_1, y_1)\) is given by:
\[ y - y_1 = m(x - x_1) \]

Since the slope \(m = 0\) and the point is \((0, 2)\), we can substitute these values into the equation:
\[ y - 2 = 0(x - 0) \]
This simplifies to:
\[ y - 2 = 0 \implies y = 2 \]

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