Question

a line passes through the points in this table.

x	0	6	9	12
y	9	5	3	1

what is the slope of the line?
write your answer as an integer or simplified fraction.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Choose two points from the table

Let's take \((x_1, y_1) = (0, 9)\) and \((x_2, y_2) = (6, 5)\).

Step3: Substitute into slope formula

\( m = \frac{5 - 9}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \)? Wait, no, let's check another pair. Take \((0,9)\) and \((9,3)\): \( m = \frac{3 - 9}{9 - 0} = \frac{-6}{9} = -\frac{2}{3} \). Or \((6,5)\) and \((12,1)\): \( m = \frac{1 - 5}{12 - 6} = \frac{-4}{6} = -\frac{2}{3} \). Wait, maybe I made a mistake. Wait, the table: x values 0,6,9,12; y values 9,5,3,1. Let's take (0,9) and (3,7)? No, the table has x=0,y=9; x=6,y=5; x=9,y=3; x=12,y=1. So the change in y: from 9 to 5 is -4, x from 0 to 6 is +6. So slope is -4/6 = -2/3? Wait, but let's check again. Wait, maybe the points are (0,9), (6,5), (9,3), (12,1). So the difference in y: 5 - 9 = -4, x: 6 - 0 = 6. So -4/6 = -2/3. Or 3 - 5 = -2, x: 9 - 6 = 3. So -2/3. Or 1 - 3 = -2, x: 12 - 9 = 3. So -2/3. So the slope is -2/3? Wait, but the problem says "Write your answer as an integer or simplified fraction." So -2/3. Wait, but maybe I misread the table. Let me check the table again. The x column: 0,6,9,12? Wait, no, the original table: x: 0, 6, 9, 12? Wait, the user's image: x has 0,6,9,12? Wait, no, maybe the x values are 0, 6, 9, 12? Wait, the y values: 9,5,3,1. So let's take (0,9) and (12,1). Then y2 - y1 = 1 - 9 = -8, x2 - x1 = 12 - 0 = 12. So -8/12 = -2/3. Yes, that's consistent. So the slope is -2/3. Wait, but maybe the table is x: 0, 6, 9, 12? Wait, no, maybe the x values are 0, 6, 9, 12? Wait, the user's image: the table has x with 0, 6, 9, 12? Wait, no, looking at the image: the x row has 0, 6, 9, 12? Wait, the first x is 0, then 6, then 9, then 12? And y row: 9,5,3,1. So yes, the slope is (1 - 9)/(12 - 0) = -8/12 = -2/3. Or (5 - 9)/(6 - 0) = -4/6 = -2/3. So the slope is -2/3.

Answer:

\(-\frac{2}{3}\)