Using slopes, find the value of x for which the points A(5, 1), B(1,-1) and C(x-4) are collinear.

Explanation

To determine the value of \( x \) for which the points \( A(5, 1) \), \( B(1, -1) \), and \( C(x, 4) \) are collinear, we can use the concept of slopes. Points are collinear if the slopes between any two pairs of points are equal.

### Step 1: Calculate the slope between points A and B

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 \( A(5, 1) \) and \( B(1, -1) \):

- \( (x_1, y_1) = (5, 1) \)
- \( (x_2, y_2) = (1, -1) \)

Substituting these values into the slope formula:

\[ m_{AB} = \frac{-1 - 1}{1 - 5} = \frac{-2}{-4} = \frac{1}{2} \]

### Step 2: Calculate the slope between points A and C

Now, we find the slope between points \( A(5, 1) \) and \( C(x, 4) \):

- \( (x_1, y_1) = (5, 1) \)
- \( (x_2, y_2) = (x, 4) \)

Using the slope formula again:

\[ m_{AC} = \frac{4 - 1}{x - 5} = \frac{3}{x - 5} \]

### Step 3: Set the slopes equal to each other

Since points \( A \), \( B \), and \( C \) are collinear, the slopes must be equal:

\[ m_{AB} = m_{AC} \]

Substituting the values we calculated:

\[ \frac{1}{2} = \frac{3}{x - 5} \]

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

To solve for \( x \), we can cross-multiply:

\[ 1 \cdot (x - 5) = 2 \cdot 3 \]

This simplifies to:

\[ x - 5 = 6 \]

Adding 5 to both sides gives:

\[ x = 11 \]

### Conclusion

The value of \( x \) for which the points \( A(5, 1) \), \( B(1, -1) \), and \( C(x, 4) \) are collinear is:

\[ \boxed{11} \]