To solve this problem, we need to determine which of the given lines could be parallel to the line [tex]\( y = 7x + 5 \)[/tex].
First, recall that for two lines to be parallel, their slopes must be equal. The equation of a line 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.
The given line has the equation [tex]\( y = 7x + 5 \)[/tex]. Here, the slope [tex]\( m \)[/tex] is 7.
We need to identify which of the given equations of lines also have a slope of 7. Let's examine each option:
1. [tex]\( y = -7x + 5 \)[/tex]: The slope here is -7. This does not match the slope of 7, so this line is not parallel to [tex]\( y = 7x + 5 \)[/tex].
2. [tex]\( y = -\frac{1}{7}x + 4 \)[/tex]: The slope here is [tex]\(-\frac{1}{7}\)[/tex]. This does not match the slope of 7, so this line is not parallel to [tex]\( y = 7x + 5 \)[/tex].
3. [tex]\( y = 5x + \frac{1}{5} \)[/tex]: The slope here is 5. This does not match the slope of 7, so this line is not parallel to [tex]\( y = 7x + 5 \)[/tex].
4. [tex]\( y = 7x - \frac{1}{5} \)[/tex]: The slope here is 7. This slope matches the slope of [tex]\( y = 7x + 5 \)[/tex], so this line is parallel to [tex]\( y = 7x + 5 \)[/tex].
Hence, the equation of line [tex]\( k \)[/tex] which is parallel to [tex]\( y = 7x + 5 \)[/tex] is [tex]\( y = 7x - \frac{1}{5} \)[/tex]. Therefore, the correct option is:
[tex]\[ \boxed{y = 7x - \frac{1}{5}} \][/tex]
