Question

homework #8 (two pages)
put points into a table if that helps!
fluency
1.) true or false: the higher the constant of proportionality, the less steep the slope.

1. the slope of a line that passes through the points (3,6) and (7,16) is

(1) \\(\frac{5}{2}\\) (3) \\(\frac{2}{5}\\)
(2) 2 (4) \\(\frac{11}{5}\\)

1. a line passes through the point (4,6). which other point could it pass through so that its slope is negative?

(1) (8,10) (3) (10,2)
(2) (5,8) (4) (9,11)

1. for each of the following sets of points, find the slope of the line that passes through them. write each of your answers in simplest form.

(a) (3, 2) and (11, 6)
(b) (2, 8) and (5, 3)

Explanation:

Question 1: True or False: The higher the constant of proportionality, the less steep the slope.

Step 1: Recall the relationship between the constant of proportionality (k) and the slope (m) in a proportional relationship (y = kx). The slope of the line y = kx is equal to k.

Step 2: Analyze the steepness of a line. A larger absolute value of the slope means a steeper line. So, if k (the constant of proportionality and the slope) is higher (larger in value), the slope is steeper, not less steep.

Step 1: Recall the slope formula. The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Identify the points. Here, \( x_1 = 3 \), \( y_1 = 6 \), \( x_2 = 7 \), \( y_2 = 16 \).

Step 3: Substitute into the formula. \( m=\frac{16 - 6}{7 - 3}=\frac{10}{4}=\frac{5}{2} \).

Step 1: Recall the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the slope to be negative, \( (y_2 - y_1) \) and \( (x_2 - x_1) \) must have opposite signs (one positive, one negative).

Step 2: Analyze each option:

• Option (1): \( (8,10) \). \( x_2 - x_1 = 8 - 4 = 4 \) (positive), \( y_2 - y_1 = 10 - 6 = 4 \) (positive). Slope \( \frac{4}{4}=1 \) (positive).
• Option (2): \( (5,8) \). \( x_2 - x_1 = 5 - 4 = 1 \) (positive), \( y_2 - y_1 = 8 - 6 = 2 \) (positive). Slope \( \frac{2}{1}=2 \) (positive).
• Option (3): \( (10,2) \). \( x_2 - x_1 = 10 - 4 = 6 \) (positive), \( y_2 - y_1 = 2 - 6 = -4 \) (negative). Slope \( \frac{-4}{6}=-\frac{2}{3} \) (negative).
• Option (4): \( (9,11) \). \( x_2 - x_1 = 9 - 4 = 5 \) (positive), \( y_2 - y_1 = 11 - 6 = 5 \) (positive). Slope \( \frac{5}{5}=1 \) (positive).

Answer:

False

Question 2: The slope of a line that passes through the points (3,6) and (7,16) is...