The line joining A (-3, 4) and B (2, -1) is parallel to the line joining C (1, -2) and D (0, x). Find x.

Answer

To find the value of \( x \) such that the line joining points \( C(1, -2) \) and \( D(0, x) \) is parallel to the line joining points \( A(-3, 4) \) and \( B(2, -1) \), we follow these steps:

### Step 1: Calculate the slope of line AB
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( A(-3, 4) \) and \( B(2, -1) \):
- \( x_1 = -3, y_1 = 4 \)
- \( x_2 = 2, y_2 = -1 \)

Substituting these values into the slope formula:
\[ m_{AB} = \frac{-1 - 4}{2 - (-3)} = \frac{-5}{2 + 3} = \frac{-5}{5} = -1 \]

### Step 2: Calculate the slope of line CD
Now, we calculate the slope of the line joining points \( C(1, -2) \) and \( D(0, x) \):
- \( x_1 = 1, y_1 = -2 \)
- \( x_2 = 0, y_2 = x \)

Using the slope formula:
\[ m_{CD} = \frac{x - (-2)}{0 - 1} = \frac{x + 2}{-1} = - (x + 2) = -x - 2 \]

### Step 3: Set the slopes equal
Since lines AB and CD are parallel, their slopes must be equal:
\[ m_{AB} = m_{CD} \]
Substituting the slopes we found:
\[ -1 = -x - 2 \]

### Step 4: Solve for \( x \)
To solve for \( x \), we rearrange the equation:
\[ -1 = -x - 2 \implies -x = -1 + 2 \implies -x = 1 \implies x = -1 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{-1} \]