Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0
What is the equation of the line passing through the point of intersection of the given lines and parallel to X-axis ?

Options

A. y + 1 = 0
B. y - 1 = 0
C. y - 2 = 0
D. y + 2 = 0

Answer

To find the equation of the line that passes through the point of intersection of the given lines and is parallel to the X-axis, we will follow these steps:

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

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

### Step 2: Solve for one variable.
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = -x - 1 \]

### Step 3: Substitute into the second equation.
Now, substitute \( y \) in Equation 2:
\[ 3x + 2(-x - 1) + 1 = 0 \]
This simplifies to:
\[ 3x - 2x - 2 + 1 = 0 \]
\[ x - 1 = 0 \]
So, we find:
\[ x = 1 \]

### Step 4: Find the corresponding \( y \) value.
Now substitute \( x = 1 \) back into the expression for \( y \):
\[ y = -1 - 1 = -2 \]
Thus, the point of intersection is:
\[ (1, -2) \]

### Step 5: Write the equation of the line parallel to the X-axis.
A line parallel to the X-axis has the form \( y = c \), where \( c \) is a constant. Since our point of intersection has a \( y \)-coordinate of \(-2\), the equation of the line is:
\[ y = -2 \]

### Final Answer:
The equation of the line passing through the point of intersection and parallel to the X-axis is:
\[ y + 2 = 0 \]

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