If point `E(5, h)` is on the line that contains `A(0,1) and B(-2, -1)`, what is the value of h?

Options

A. `-1`
B. `0`
C. `1`
D. `6`

Answer

To find the value of \( h \) for the point \( E(5, h) \) that lies on the line containing points \( A(0, 1) \) and \( B(-2, -1) \), we will follow these steps:

### Step 1: Find the slope of line AB
The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( A(0, 1) \) and \( B(-2, -1) \):
- \( (x_1, y_1) = (0, 1) \)
- \( (x_2, y_2) = (-2, -1) \)

Substituting the values:
\[ m = \frac{-1 - 1}{-2 - 0} = \frac{-2}{-2} = 1 \]

### Step 2: Use the point-slope form to find the equation of line AB
The point-slope form of the equation of a line is:
\[ y - y_1 = m(x - x_1) \]
Using point \( A(0, 1) \) and the slope \( m = 1 \):
\[ y - 1 = 1(x - 0) \]
This simplifies to:
\[ y - 1 = x \]
Rearranging gives:
\[ y = x + 1 \]

### Step 3: Substitute point E into the equation
Now, we substitute the coordinates of point \( E(5, h) \) into the equation \( y = x + 1 \):
\[ h = 5 + 1 \]
Thus:
\[ h = 6 \]

### Final Answer
The value of \( h \) is \( 6 \).

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