To write the equation of a line parallel to the given line [tex]\( EF \)[/tex], we need to understand the properties of parallel lines. Parallel lines have the same slope.
Given:
- The equation of line [tex]\( EF \)[/tex] is [tex]\( y = 2x + 1 \)[/tex].
From this equation, we can see that the slope [tex]\( m \)[/tex] of line [tex]\( EF \)[/tex] is 2.
We need to find the equation of a line that is parallel to line [tex]\( EF \)[/tex] and passes through the point [tex]\( (0, 2) \)[/tex].
Since parallel lines have the same slope, the slope of our desired line is also 2.
The general form of a line's equation in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Substituting the slope [tex]\( m = 2 \)[/tex], we get:
[tex]\[ y = 2x + b \][/tex]
Now, we need to determine [tex]\( b \)[/tex] using the point [tex]\( (0, 2) \)[/tex].
Substitute the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values from the point [tex]\( (0, 2) \)[/tex] into the equation:
[tex]\[ 2 = 2(0) + b \][/tex]
[tex]\[ 2 = b \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is 2.
Therefore, the equation of the line parallel to [tex]\( EF \)[/tex] and passing through the point [tex]\( (0, 2) \)[/tex] is:
[tex]\[ y = 2x + 2 \][/tex]
Among the given choices, this corresponds to the second option:
[tex]\[ y = 2x + 2 \][/tex]
