To find the equation of a line parallel to the given line [tex]\( EF \mathpunct{:} y = 2x + 1 \)[/tex] that passes through the point [tex]\( (0, 2) \)[/tex], follow these steps:
1. Identify the slope of the given line: The equation of line [tex]\( EF \)[/tex] is given in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For [tex]\( y = 2x + 1 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
2. Parallel lines have the same slope: Since we are looking for a line parallel to [tex]\( EF \)[/tex], the slope of our new line must also be [tex]\( 2 \)[/tex].
3. Use the point-slope form of the line equation: The point-slope form of a line equation with a point [tex]\( (x_1, y_1) \)[/tex] and slope [tex]\( m \)[/tex] is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex].
4. Substitute the given point and the slope into the point-slope form: The given point is [tex]\( (0, 2) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex]. Substitute these values into the point-slope form:
[tex]\[
y - 2 = 2(x - 0)
\][/tex]
5. Simplify the equation to get the slope-intercept form:
[tex]\[
y - 2 = 2x
\][/tex]
[tex]\[
y = 2x + 2
\][/tex]
Now, we have the equation of the line which is parallel to [tex]\( EF \)[/tex] and passes through the point [tex]\( (0, 2) \)[/tex] as [tex]\( y = 2x + 2 \)[/tex].
Thus, the correct answer is:
[tex]\[ y = 2x + 2 \][/tex]
