Question

A line passes through the points (4,6) and (6,2).


Select Yes or No to tell whether each equation describes this line.

Equation Yes No
y−6=−2(x−4)


y−4=−2(x−6)


​y−2=−2(x+6)​


y−2=−2(x−6)

Answer 1

To determine whether each equation describes the line passing through the points (4,6) and (6,2), we first need to find the slope of the line through these points.

The slope \( m \) is calculated as follows:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 6}{6 - 4} = \frac{-4}{2} = -2
\]

Now, we have the slope \( m = -2 \). We can use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

Using the point (4,6):

\[
y - 6 = -2(x - 4)
\]

Let's analyze each equation:

1. **Equation: \( y - 6 = -2(x - 4) \)**

This is the correct equation we derived by using the point (4,6) with the slope of -2.

- **Answer: Yes**

2. **Equation: \( y - 4 = -2(x - 6) \)**

If we simplify this equation using point (6,2):

\[
y - 2 = -2(x - 6),
\]
which gives us another form using the point (6,2) and also has the correct slope.

- **Answer: Yes**

3. **Equation: \( y - 2 = -2(x + 6) \)**

This equation uses the point (6,2) but incorrectly refers to point (−6). Thus, it is not a correct representation of the line.

- **Answer: No**

4. **Equation: \( y - 2 = -2(x - 6) \)**

This is another way to derive the equation using the point (6,2), having the slope -2.

- **Answer: Yes**

Now, we summarize the answers:

- \( y - 6 = -2(x - 4) \): **Yes**
- \( y - 4 = -2(x - 6) \): **Yes**
- \( y - 2 = -2(x + 6) \): **No**
- \( y - 2 = -2(x - 6) \): **Yes**