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. If three points are collinear, the slope between any two pairs of points must be equal.

### Step-by-Step Solution:

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

2. **Calculate the Slope of AB**:
The slope of line segment \( AB \) is given by the formula:
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-1)}{2 - x} = \frac{1 + 1}{2 - x} = \frac{2}{2 - x} \]

3. **Calculate the Slope of BC**:
The slope of line segment \( BC \) is:
\[ \text{slope of } BC = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2 \]

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

5. **Cross Multiply**:
To eliminate the fraction, we can cross multiply:
\[ 2 = 2(2 - x) \]

6. **Distribute and Simplify**:
Distributing the right side gives:
\[ 2 = 4 - 2x \]

7. **Rearrange the Equation**:
Rearranging the equation to isolate \( x \):
\[ 2x = 4 - 2 \]
\[ 2x = 2 \]

8. **Solve for x**:
Dividing both sides by 2 gives:
\[ x = 1 \]

### Conclusion:
Thus, the value of \( x \) for which the points are collinear is:
\[ \boxed{1} \]