Question

11、find the slope of the line that passes through all of the points on the table.
x | y
1 | 4
4 | 12
7 | 20
10 | 28

Explanation:

Step1: Recall the slope formula

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

Step2: Choose two points from the table

Let's take the first two points \((1, 4)\) and \((4, 12)\). Here, \( x_1 = 1 \), \( y_1 = 4 \), \( x_2 = 4 \), \( y_2 = 12 \).

Step3: Calculate the slope

Substitute the values into the slope formula:
\( m=\frac{12 - 4}{4 - 1}=\frac{8}{3} \)? Wait, no, wait. Wait, let's check another pair to confirm. Let's take \((4, 12)\) and \((7, 20)\). Then \( x_1 = 4 \), \( y_1 = 12 \), \( x_2 = 7 \), \( y_2 = 20 \). Then \( m=\frac{20 - 12}{7 - 4}=\frac{8}{3} \)? Wait, no, wait \( 20 - 12 = 8 \), \( 7 - 4 = 3 \), so \( \frac{8}{3} \)? Wait, but wait, let's check \((1,4)\) and \((7,20)\). \( y_2 - y_1 = 20 - 4 = 16 \), \( x_2 - x_1 = 7 - 1 = 6 \), \( \frac{16}{6}=\frac{8}{3} \). Wait, but wait, maybe I made a mistake. Wait, let's check the first two points again. \( (1,4) \) and \( (4,12) \): \( 12 - 4 = 8 \), \( 4 - 1 = 3 \), so \( \frac{8}{3} \approx 2.666... \). Wait, but let's check \((4,12)\) and \((10,28)\). \( 28 - 12 = 16 \), \( 10 - 4 = 6 \), \( \frac{16}{6}=\frac{8}{3} \). Oh, so the slope is consistent. Wait, but wait, maybe I miscalculated. Wait, let's do it again. The slope formula is \( m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take \((1,4)\) and \((4,12)\): \( \Delta y = 12 - 4 = 8 \), \( \Delta x = 4 - 1 = 3 \), so \( m = \frac{8}{3} \). Wait, but let's check with \((7,20)\) and \((10,28)\): \( \Delta y = 28 - 20 = 8 \), \( \Delta x = 10 - 7 = 3 \), so \( m = \frac{8}{3} \). So the slope is \( \frac{8}{3} \)? Wait, but maybe I made a mistake. Wait, let's check the first point \((1,4)\) and \((10,28)\): \( \Delta y = 28 - 4 = 24 \), \( \Delta x = 10 - 1 = 9 \), \( \frac{24}{9}=\frac{8}{3} \). Yes, so the slope is \( \frac{8}{3} \). Wait, but maybe the problem is designed to have a slope of \( \frac{8}{3} \). Wait, but let me check again. Wait, maybe I misread the table. The table is:

x | y

1 | 4

4 | 12

7 | 20

10 | 28

So the x-values are increasing by 3 each time (1 to 4 is +3, 4 to 7 is +3, 7 to 10 is +3), and the y-values are increasing by 8 each time (4 to 12 is +8, 12 to 20 is +8, 20 to 28 is +8). So the slope is \( \frac{\Delta y}{\Delta x}=\frac{8}{3} \). Yes, that's correct. So the slope is \( \frac{8}{3} \).

Answer:

\(\frac{8}{3}\)