How do you tell whether the lines through the given points are parallel, perpendicular, or neither: (1,0), (7,4)?

To determine whether the lines through the given points are parallel, perpendicular, or neither, we can calculate the slope of each line using the formula: ( m = \frac{{y2 - y1}}{{x2 - x1}} ). If the slopes of the lines are equal, they are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. For the points (1,0) and (7,4), the slope is ( m = \frac{{4 - 0}}{{7 - 1}} = \frac{4}{6} = \frac{2}{3} ). Therefore, the slope of the line passing through these points is ( \frac{2}{3} ). Since we only have one line, we cannot determine if it's parallel or perpendicular to another line without knowing the equation of the other line. Hence, we cannot definitively classify the relationship between the lines as parallel, perpendicular, or neither based solely on the given points.