Find the equation of the line through the point of intersection of 2x - 3y + 1 = 0 and x + y - 2 = 0 which is parallel to the y-axis.

Options

A. x = 1
B. 8x = 7
C. x + 3 = 0
D. x = 6

Answer

To find the equation of the line through the point of intersection of the two given lines, which is parallel to the y-axis, we can follow these steps:

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

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

### Step 2: Solve the equations
From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = 2 - y \]

Now, substitute this expression for \( x \) into Equation 1:
\[ 2(2 - y) - 3y + 1 = 0 \]
Expanding this gives:
\[ 4 - 2y - 3y + 1 = 0 \]
Combining like terms:
\[ 4 + 1 - 5y = 0 \implies 5 - 5y = 0 \implies -5y = -5 \implies y = 1 \]

### Step 3: Substitute \( y \) back to find \( x \)
Now that we have \( y = 1 \), substitute this back into Equation 2 to find \( x \):
\[ x + 1 - 2 = 0 \implies x - 1 = 0 \implies x = 1 \]

Thus, the point of intersection is \( (1, 1) \).

### Step 4: Find the equation of the line parallel to the y-axis
A line that is parallel to the y-axis has a constant \( x \) value. Since the line we are looking for passes through the point of intersection \( (1, 1) \), the equation of the line will be:
\[ x = 1 \]

### Final Answer
The equation of the line through the point of intersection of the two lines, which is parallel to the y-axis, is:
\[ \boxed{x = 1} \]