To determine which equation could represent the line [tex]\( k \)[/tex] that is parallel to the line with the equation [tex]\( y = 7x + 5 \)[/tex], we need to understand the properties of parallel lines in the [tex]\( xy \)[/tex]-plane.
1. Parallel lines have equal slopes. Therefore, the slope of line [tex]\( k \)[/tex] must match the slope of the given line [tex]\( y = 7x + 5 \)[/tex].
2. The given line has the equation [tex]\( y = 7x + 5 \)[/tex]. In the slope-intercept form [tex]\( y = mx + b \)[/tex], the coefficient of [tex]\( x \)[/tex] (m) represents the slope. Here, the slope (m) is [tex]\( 7 \)[/tex].
Since line [tex]\( k \)[/tex] must be parallel to this line, the slope of line [tex]\( k \)[/tex] must also be [tex]\( 7 \)[/tex]. We now look for the equation among the given options that has a slope of [tex]\( 7 \)[/tex].
- The first option [tex]\( y = -7x + 5 \)[/tex] has a slope of [tex]\( -7 \)[/tex]. This is not parallel to [tex]\( y = 7x + 5 \)[/tex].
- The second option [tex]\( y = -\frac{1}{7}x + 4 \)[/tex] has a slope of [tex]\( -\frac{1}{7} \)[/tex]. This is not parallel to [tex]\( y = 7x + 5 \)[/tex].
- The third option [tex]\( y = 5x + \frac{1}{5} \)[/tex] has a slope of [tex]\( 5 \)[/tex]. This is not parallel to [tex]\( y = 7x + 5 \)[/tex].
- The fourth option [tex]\( y = 7x - \frac{1}{5} \)[/tex] has a slope of [tex]\( 7 \)[/tex]. This matches the slope of the line [tex]\( y = 7x + 5 \)[/tex].
Thus, the equation of the line [tex]\( k \)[/tex] that is parallel to the line [tex]\( y = 7x + 5 \)[/tex] is:
[tex]\[ y = 7x - \frac{1}{5} \][/tex]
Therefore, the correct answer is:
[tex]\[ y = 7x - \frac{1}{5} \][/tex]
