Find the equation of a line which passes through the intersection of lines `2x+y-1=0` and `x-3y+1=0` and parallel to `x`-axis.

Answer

To find the equation of a line that passes through the intersection of the lines \(2x + y - 1 = 0\) and \(x - 3y + 1 = 0\) and is parallel to the x-axis, we can follow these steps:

### Step 1: Find the intersection of the two lines
We have two equations:
1. \(2x + y - 1 = 0\) (Equation 1)
2. \(x - 3y + 1 = 0\) (Equation 2)

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

### Step 2: Substitute \(y\) into the second equation
Now, substitute \(y = 1 - 2x\) into Equation 2:
\[ x - 3(1 - 2x) + 1 = 0 \]
Expanding this gives:
\[ x - 3 + 6x + 1 = 0 \]
Combining like terms:
\[ 7x - 2 = 0 \]

### Step 3: Solve for \(x\)
Now, solve for \(x\):
\[ 7x = 2 \implies x = \frac{2}{7} \]

### Step 4: Find the corresponding \(y\) value
Substituting \(x = \frac{2}{7}\) back into the equation for \(y\):
\[ y = 1 - 2\left(\frac{2}{7}\right) = 1 - \frac{4}{7} = \frac{3}{7} \]

### Step 5: Write the coordinates of the intersection point
The intersection point of the two lines is:
\[ \left(\frac{2}{7}, \frac{3}{7}\right) \]

### Step 6: Determine the equation of the line parallel to the x-axis
A line that is parallel to the x-axis has a constant \(y\) value. Since our intersection point has a \(y\) value of \(\frac{3}{7}\), the equation of the line is:
\[ y = \frac{3}{7} \]

### Final Answer
Thus, the equation of the line that passes through the intersection of the two lines and is parallel to the x-axis is:
\[ \boxed{y = \frac{3}{7}} \]