To find the equation of a line parallel to [tex]\( y = 2x + 2 \)[/tex] that passes through the point [tex]\( (3, -1) \)[/tex], we follow these steps:
1. Identify the slope of the parallel line:
Since parallel lines have the same slope, the slope [tex]\( m \)[/tex] of our line will be the same as the slope of the given line [tex]\( y = 2x + 2 \)[/tex]. The slope of this line is [tex]\( 2 \)[/tex].
2. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, [tex]\(m = 2\)[/tex], [tex]\(x_1 = 3\)[/tex], and [tex]\(y_1 = -1\)[/tex].
3. Substitute the given point and slope into the point-slope form:
[tex]\[
y - (-1) = 2 (x - 3)
\][/tex]
Simplifying the left-hand side:
[tex]\[
y + 1 = 2 (x - 3)
\][/tex]
4. Identify the values for the point-slope form representation:
We need to express the equation in the form [tex]\( y - [?] = 2 (x - ?) \)[/tex]:
- The value that should replace [tex]\([?]\)[/tex] in [tex]\( y - [?] = \ldots \)[/tex] is [tex]\( -1 \)[/tex].
Therefore, the value that belongs in the green box is [tex]\(\boxed{-1}\)[/tex].
