Question

Which equation is a point slope form equation for line AB
?

Responses

y+6=−2(x−1)
y plus 6 equals negative 2 left parenthesis x minus 1 right parenthesis

y+2=−2(x−5)
y plus 2 equals negative 2 left parenthesis x minus 5 right parenthesis

y+1=−2(x−6)
y minus 1 equals negative 2 left parenthesis x minus 6 right parenthesis

y+5=−2(x−2)
y plus 5 equals negative 2 left parenthesis x minus 2 right parenthesis
A graph with a line running through point A, with coordinates (1, 6), and point B, with coordinates (5, -2)

Answer 1

To determine the correct point-slope form equation for the line passing through the points A (1, 6) and B (5, -2), we first need to confirm the slope and a point on the line.

1. **Finding the slope (m)**:
The slope \( m \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, we'll take point A as \( (x_1, y_1) = (1, 6) \) and point B as \( (x_2, y_2) = (5, -2) \).

Substituting these values, we have:
\[
m = \frac{-2 - 6}{5 - 1} = \frac{-8}{4} = -2
\]

2. **Using the point-slope form**:
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
We can use either point A or point B to write the equation. Let's use point A (1, 6):

Substituting \( m = -2 \), \( x_1 = 1 \), and \( y_1 = 6 \):
\[
y - 6 = -2(x - 1)
\]

3. **Rearranging**:
To match the responses, we can rearrange this equation:
\[
y + 6 = -2(x - 1)
\]

Now, let's compare the derived equation with the given options:

- \( y + 6 = -2(x - 1) \) (This matches our derived equation.)
- \( y + 2 = -2(x - 5) \)
- \( y + 1 = -2(x - 6) \)
- \( y + 5 = -2(x - 2) \)

The correct equation in point-slope form for line AB is:

**\( y + 6 = -2(x - 1) \)**.