If the points `(-1,y)`, `(1,2)` and `(5,4)` are collinear, find the value of `y`.

Answer

To find the value of \( y \) such that the points \((-1,y)\), \((1,2)\), and \((5,4)\) are collinear, we can follow these steps:

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

### Step 2: Calculate the slope between points \((1,2)\) and \((5,4)\)
Using the slope formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \((1,2)\) and \((5,4)\):
- \(x_1 = 1\), \(y_1 = 2\)
- \(x_2 = 5\), \(y_2 = 4\)

Substituting these values into the slope formula:
\[ \text{slope}_{BC} = \frac{4 - 2}{5 - 1} = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Calculate the slope between points \((-1,y)\) and \((1,2)\)
Using the same slope formula:
For points \((-1,y)\) and \((1,2)\):
- \(x_1 = -1\), \(y_1 = y\)
- \(x_2 = 1\), \(y_2 = 2\)

Substituting these values into the slope formula:
\[ \text{slope}_{AB} = \frac{2 - y}{1 - (-1)} = \frac{2 - y}{1 + 1} = \frac{2 - y}{2} \]

### Step 4: Set the slopes equal to each other
Since the points are collinear, we can set the slopes equal:
\[ \frac{2 - y}{2} = \frac{1}{2} \]

### Step 5: Solve for \( y \)
To eliminate the fraction, multiply both sides by 2:
\[ 2 - y = 1 \]
Now, isolate \( y \):
\[ -y = 1 - 2 \]
\[ -y = -1 \]
\[ y = 1 \]

### Conclusion
The value of \( y \) is \( 1 \).

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