Calculate the slope of the line passing through each pair of points below. Homework Help a \( \begin{array}{ll}\text { a. }(1,2) \text { and }(4,-1) & \text { b. }(7,3) \text { and }(5,4) \\ \text { c. }(-6,8) \text { and }(-8,5) & \text { d. }(55,67) \text { and }(50,68) \\ \text { e. Azizah got } 1 \text { for the slope of the line through points }(1,2) \text { and }(4,-1) \text {. Explain to her the mistake she made and how } \\ \text { to calculate the slope correctly. }\end{array} \)

To calculate the slope of a line passing through two points, you can use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) is the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Let's go through each pair of points and calculate the slope: a. Points: \( (1,2) \) and \( (4,-1) \) \[ m = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1 \] b. Points: \( (7,3) \) and \( (5,4) \) \[ m = \frac{4 - 3}{5 - 7} = \frac{1}{-2} = -\frac{1}{2} \] c. Points: \( (-6,8) \) and \( (-8,5) \) \[ m = \frac{5 - 8}{-8 - (-6)} = \frac{-3}{-2} = \frac{3}{2} \] d. Points: \( (55,67) \) and \( (50,68) \) \[ m = \frac{68 - 67}{50 - 55} = \frac{1}{-5} = -\frac{1}{5} \] e. Azizah's calculation: Azizah calculated the slope of the line through points \( (1,2) \) and \( (4,-1) \) as follows: \[ m = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1 \] Azizah's mistake: Azizah made a mistake in her calculation. She correctly identified the points and the formula for slope, but she did not perform the subtraction correctly. She subtracted the x-coordinates and the y-coordinates in the wrong order. The correct calculation should be: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1 \] So, Azizah's final answer is correct, and she did not make a mistake in her calculation. The slope of the line through points \( (1,2) \) and \( (4,-1) \) is indeed \(-1\).