63. ((1)/(2),(1)/(3)) and ((3)/(2),(5)/(3))

To determine the relationship between the two points \((\frac{1}{2}, \frac{1}{3})\) and \((\frac{3}{2}, \frac{5}{3})\), we can calculate the slope of the line that passes through these points.<br /><br />The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:<br /><br />\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]<br /><br />Substituting the given points \((\frac{1}{2}, \frac{1}{3})\) and \((\frac{3}{2}, \frac{5}{3})\):<br /><br />\[ m = \frac{\frac{5}{3} - \frac{1}{3}}{\frac{3}{2} - \frac{1}{2}} \]<br /><br />Simplify the numerator and the denominator:<br /><br />\[ m = \frac{\frac{5}{3} - \frac{1}{3}}{\frac{3}{2} - \frac{1}{2}} = \frac{\frac{4}{3}}{\frac{2}{2}} = \frac{\frac{4}{3}}{1} = \frac{4}{3} \]<br /><br />So, the slope of the line passing through the points \((\frac{1}{2}, \frac{1}{3})\) and \((\frac{3}{2}, \frac{5}{3})\) is \(\frac{4}{3}\).<br /><br />This means that the line has a positive slope and rises 4 units for every 3 units it moves horizontally.