Find the value of x for which the points `(x, -1)` (2,1) and (4, 5) are collinear.

Answer

To find the value of \( x \) for which the points \( (x, -1) \), \( (2, 1) \), and \( (4, 5) \) are collinear, we can use the concept of slopes. For three points to be collinear, the slope between any two pairs of points must be equal.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let point A be \( (x, -1) \)
- Let point B be \( (2, 1) \)
- Let point C be \( (4, 5) \)

2. **Calculate the Slope of Line AB**:
- The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
- For points A and B:
\[ \text{slope of AB} = \frac{1 - (-1)}{2 - x} = \frac{2}{2 - x} \]

3. **Calculate the Slope of Line BC**:
- For points B and C:
\[ \text{slope of BC} = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2 \]

4. **Set the Slopes Equal**:
- Since the points are collinear, we set the slopes equal to each other:
\[ \frac{2}{2 - x} = 2 \]

5. **Solve for \( x \)**:
- Cross-multiply to eliminate the fraction:
\[ 2 = 2(2 - x) \]
- Distribute the 2 on the right side:
\[ 2 = 4 - 2x \]
- Rearrange the equation to isolate \( x \):
\[ 2x = 4 - 2 \]
\[ 2x = 2 \]
\[ x = 1 \]

6. **Conclusion**:
- The value of \( x \) for which the points are collinear is \( x = 1 \).