If the points `P(1,5)`, `Q(-1,1)` and `R(4,y)` are collinear, find the value of `y`.

Options

A. `y=-12`
B. `y=12`
C. `y=11`
D. `y=-11`

Explanation

To find the value of \( y \) such that the points \( P(1,5) \), \( Q(-1,1) \), and \( R(4,y) \) are collinear, we can follow these steps:

### Step 1: Understand the concept of collinearity
Collinear points lie on the same straight line. This means that the slope between any two pairs of points must be the same.

### Step 2: Calculate the slope between points \( P \) and \( Q \)
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 \( P(1, 5) \) and \( Q(-1, 1) \):
\[ m_{PQ} = \frac{1 - 5}{-1 - 1} = \frac{-4}{-2} = 2 \]

### Step 3: Calculate the slope between points \( Q \) and \( R \)
Using the same slope formula for points \( Q(-1, 1) \) and \( R(4, y) \):
\[ m_{QR} = \frac{y - 1}{4 - (-1)} = \frac{y - 1}{4 + 1} = \frac{y - 1}{5} \]

### Step 4: Set the slopes equal to each other
Since the points are collinear, the slopes must be equal:
\[ m_{PQ} = m_{QR} \]
Thus, we have:
\[ 2 = \frac{y - 1}{5} \]

### Step 5: Solve for \( y \)
To solve for \( y \), we can cross-multiply:
\[ 2 \cdot 5 = y - 1 \]
\[ 10 = y - 1 \]
Now, add 1 to both sides:
\[ y = 10 + 1 = 11 \]

### Conclusion
The value of \( y \) is \( 11 \).

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