Find the value of k for which the points A(-1, 3), B(2, k) and C(5, -1) are collinear.

Answer

To find the value of \( k \) for which the points \( A(-1, 3) \), \( B(2, k) \), and \( C(5, -1) \) are collinear, we can use the formula for the area of a triangle formed by three points in a coordinate plane. The area of the triangle formed by points \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

If the points are collinear, the area will be zero. Therefore, we set the area formula equal to zero:

\[ \frac{1}{2} \left| -1(k - (-1)) + 2(-1 - 3) + 5(3 - k) \right| = 0 \]

Now, we can simplify this step by step.

### Step 1: Substitute the coordinates into the area formula

Substituting the coordinates of points \( A \), \( B \), and \( C \):

\[ \frac{1}{2} \left| -1(k + 1) + 2(-4) + 5(3 - k) \right| = 0 \]

### Step 2: Simplify the expression inside the absolute value

Calculating each term:

\[ -1(k + 1) = -k - 1 \]
\[ 2(-4) = -8 \]
\[ 5(3 - k) = 15 - 5k \]

Now, combine these:

\[ -k - 1 - 8 + 15 - 5k = -6k + 6 \]

### Step 3: Set the equation equal to zero

Since the area is zero:

\[ \frac{1}{2} \left| -6k + 6 \right| = 0 \]

This implies:

\[ \left| -6k + 6 \right| = 0 \]

### Step 4: Solve for \( k \)

The absolute value equals zero when the expression inside is zero:

\[ -6k + 6 = 0 \]

Rearranging gives:

\[ -6k = -6 \]

Dividing both sides by -6:

\[ k = 1 \]

### Conclusion

Thus, the value of \( k \) for which the points \( A \), \( B \), and \( C \) are collinear is:

\[ \boxed{1} \]