Find the slope of the line that r ((1)/(2),-(4)/(3)) and ((7)/(15),(4)/(3)) Find the value of 'r'so that t

To find the slope of the line passing through the points \((\frac{1}{2}, -\frac{4}{3})\) and \((\frac{7}{15}, \frac{4}{3})\), we use the slope formula:<br /><br />\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]<br /><br />Here, \((x_1, y_1) = (\frac{1}{2}, -\frac{4}{3})\) and \((x_2, y_2) = (\frac{7}{15}, \frac{4}{3})\).<br /><br />Substitute these values into the formula:<br /><br />\[ m = \frac{\frac{4}{3} - (-\frac{4}{3})}{\frac{7}{15} - \frac{1}{2}} \]<br /><br />Simplify the numerator:<br /><br />\[ \frac{4}{3} - (-\frac{4}{3}) = \frac{4}{3} + \frac{4}{3} = \frac{8}{3} \]<br /><br />Now, simplify the denominator:<br /><br />\[ \frac{7}{15} - \frac{1}{2} \]<br /><br />To subtract these fractions, find a common denominator. The least common multiple of 15 and 2 is 30.<br /><br />Convert \(\frac{7}{15}\) to a fraction with a denominator of 30:<br /><br />\[ \frac{7}{15} = \frac{7 \times 2}{15 \times 2} = \frac{14}{30} \]<br /><br />Convert \(\frac{1}{2}\) to a fraction with a denominator of 30:<br /><br />\[ \frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30} \]<br /><br />Now subtract the fractions:<br /><br />\[ \frac{14}{30} - \frac{15}{30} = \frac{14 - 15}{30} = \frac{-1}{30} \]<br /><br />So the slope \( m \) is:<br /><br />\[ m = \frac{\frac{8}{3}}{\frac{-1}{30}} = \frac{8}{3} \times \frac{30}{-1} = \frac{8 \times 30}{3 \times -1} = \frac{240}{-3} = -80 \]<br /><br />Therefore, the slope of the line passing through the given points is \(-80\).