If points A(-3, 12), B(7,6) and C(x, 9) are collinear, then the value of x is……….

Answer

To find the value of \( x \) such that the points \( A(-3, 12) \), \( B(7, 6) \), and \( C(x, 9) \) are collinear, we can use the formula for the area of a triangle formed by three points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(-3, 12) \) be \( (x_1, y_1) \)
- Let \( B(7, 6) \) be \( (x_2, y_2) \)
- Let \( C(x, 9) \) be \( (x_3, y_3) \)

2. **Use the Area Formula**:
The area \( A \) of the triangle formed by the points \( A \), \( B \), and \( C \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Since the points are collinear, the area will be zero:
\[ 0 = \frac{1}{2} \left| -3(6 - 9) + 7(9 - 12) + x(12 - 6) \right| \]

3. **Simplify the Equation**:
\[ 0 = \frac{1}{2} \left| -3(-3) + 7(-3) + x(6) \right| \]
\[ 0 = \frac{1}{2} \left| 9 - 21 + 6x \right| \]
\[ 0 = \frac{1}{2} \left| 6x - 12 \right| \]

4. **Eliminate the Absolute Value**:
Since the area is zero, we can set the expression inside the absolute value to zero:
\[ 6x - 12 = 0 \]

5. **Solve for \( x \)**:
\[ 6x = 12 \]
\[ x = \frac{12}{6} = 2 \]

Thus, the value of \( x \) is \( 2 \).