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 that passes through the points (4, 6) and (6, 2), we can first find the slope of the line defined by these two points.

The slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 6}{6 - 4} = \frac{-4}{2} = -2
\]

Now that we know the slope is -2, we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]

Using the point (4, 6):
\[
y - 6 = -2(x - 4) \quad \text{(This matches the first equation)}
\]

Now using the point (6, 2):
\[
y - 2 = -2(x - 6) \quad \text{(This matches the second equation)}
\]

Let's analyze the other given equations:

1. \( y - 2 = -2(x + 6) \)
- Substitute \( x = 6 \):
\[
y - 2 = -2(6 + 6) = -2(12) = -24 \quad \Rightarrow \quad y = -22 \quad (\text{Not on the line})
\]
- **Answer: No**

2. \( y - 2 = -2(x - 6) \)
- Substitute \( x = 6 \):
\[
y - 2 = -2(6 - 6) = -2(0) = 0 \quad \Rightarrow \quad y = 2 \quad (\text{Is on the line})
\]
- **Answer: Yes**

Now we can 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**

So the final answers are:

1. Yes
2. Yes
3. No
4. Yes