The final conclusions for each equation are based on the analysis of their slopes in comparison to the slope of line a.
To determine whether the given equations are parallel, perpendicular, or neither to the line y=-7+1, we can compare
their slopes
The given line y = -7x + 1 is in the form y = mx + b, where m is the slope of the line. In this case, slope m is -7
Parallel lines:
Two lines are parallel if they have the same slope. If the slope of a line is m, then any line parallel to it will have a slope m. So, for two lines to be parallel, the slopes must be equal.
Perpendicular lines:
Two lines are perpendicular if the product of their slopes is -1. If the slope of one line is
m, then the slope of a line perpendicular to it will be -1/m
Now, let's compare the given equations with the line
y= -7x+1
Equation y=7x−5:
The slope of this line is 7. Since it is the negative reciprocal of -7, these two lines are perpendicular.
Equation y - 7x+3
The slope of this line is 7, which is the same as the slope of the given line y= -7x +1 Therefore, these two lines are parallel.
Equation y - 1/4x +4
The slope of this line is 1/4 which is not the negative reciprocal of -7, nor is it equal to -7. Therefore, these two lines are neither parallel nor perpendicular.