To examine which equations in point-slope form represent the given line, we need to analyze the points provided in the table and determine the correct mathematical representation for the line passing through them.
The points given are:
[tex]\[
(-3, -6), \quad (1, 2), \quad (2, 4)
\][/tex]
First, let's find the slope [tex]\( m \)[/tex] of the line using the first two points [tex]\((-3, -6)\)[/tex] and [tex]\( (1, 2) \)[/tex].
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values:
[tex]\[
m = \frac{2 - (-6)}{1 - (-3)} = \frac{2 + 6}{1 + 3} = \frac{8}{4} = 2
\][/tex]
The slope of the line is [tex]\( m = 2 \)[/tex].
Next, we write the equations of the line in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex].
Let's consider each point and form the equations.
### Using the point [tex]\((-3, -6)\)[/tex]:
[tex]\[
y - (-6) = 2(x - (-3)) \implies y + 6 = 2(x + 3)
\][/tex]
This equation is:
[tex]\[ y + 6 = 2(x + 3) \][/tex]
### Using the point [tex]\( (1, 2) \)[/tex]:
[tex]\[
y - 2 = 2(x - 1)
\][/tex]
This equation is:
[tex]\[ y - 2 = 2(x - 1) \][/tex]
### Using the point [tex]\( (2, 4) \)[/tex]:
[tex]\[
y - 4 = 2(x - 2)
\][/tex]
This equation is:
[tex]\[ y - 4 = 2(x - 2) \][/tex]
Now, we check these equations against the given options:
1. [tex]\( y + 4 = \frac{1}{2} (x + 2) \)[/tex]
2. [tex]\( y - 4 = 2 (x - 2) \)[/tex]
3. [tex]\( y - 6 = \frac{1}{2} (x - 3) \)[/tex]
4. [tex]\( y + 2 = \frac{1}{2} (x + 1) \)[/tex]
5. [tex]\( y + 6 = 2 (x + 3) \)[/tex]
6. [tex]\( y - 2 = 2 (x - 1) \)[/tex]
From our derived equations:
- [tex]\( y + 6 = 2(x + 3) \)[/tex] is part of option 5.
- [tex]\( y - 2 = 2(x - 1) \)[/tex] is part of option 6.
- [tex]\( y - 4 = 2(x - 2) \)[/tex] is part of option 2.
Thus, the correct equations representing the line based on the points given are:
[tex]\[
y - 4 = 2(x - 2) \quad \text{(Option 2)}, \quad
y + 6 = 2(x + 3) \quad \text{(Option 5)}, \quad
y - 2 = 2(x - 1) \quad \text{(Option 6)}
\][/tex]
Therefore, after examining the points and the given options, none of the options that use [tex]\(\frac{1}{2}\)[/tex] as a slope are suitable, and the correct options are:
[tex]\[ \boxed{2, 5, 6} \][/tex]
